I was wondering how atheists determine if a particular action is moral or immoral? Is incest between consenting adults moral? Is sex with multiple partners without marriage, prostitution by one's personal wish, pornography, robbing rich guys who made wealth through illegal means immoral? Is the concept of marriage moral? If there are no moral standard, how could we ever determine the morality of an action.
Atheists generally follow the Golden Rule . A formal argument can be made for any self-consistent naturalistic moral system to obey the golden rule . Individual value tend to value in themselves 1) being alive/healthy and 2) having a happy/meaningful life. Therefore, by the golden rule, all atheists should value well-being for all people, no matter what else they think about philosophy and morality. Sam Harris is very effective in explaining why all morality is about wellbeing and why moral questions are often reducible to empirical questions - here is a talk  and a book .
Atheists are generally consequentialists  as well. When determining whether or not an action is moral, one thinks about how that action impacts the life and well-being of others. So murder is wrong because it takes away someone's life. Killing in self-defense might be okay, because you save a life (your own, possibly others) at the same time you take a life . Failing to take an opportunity to save someone's life, for example by failing to preventing a blind person from walking into oncoming traffic, would also be wrong because it results in a loss of life . Approving the design of car with a high probability of dangerous failures  is also wrong because it is likely to result in the loss of life or injury. Injury, psychological trauma, illness, sadness, a lack of purpose are also to be avoided while the health, happiness, and meaningful experiences are to be valued. (The difference between happiness and a sense of meaning is explained a bit here , obviously there is nothing about atheism which precludes the possibility of a meaningful existence.)
If you think about consequentialism a bit, you realize its an ethical code that becomes quite complex to compute. At each moment we can chose between a nearly infinite number of actions, for every action there are a nearly infinite number of possible consequences, each of those consequences lead into a nearly infinite number of other possible consequences, etc. In practice, consequentialists tend to give greater weight to the known consequences (like in the examples I've just given) . We also employ strategies that generally lead to good results: honesty , hard-work , diversity of opinion, equality, rule of law , personal freedom and helping the less fortunate. When these values conflict, we turn to the available evidence to help decide what course of action is mostly likely to contribute to minimize harm. In order to most accurately do the right thing, we atheists are also often rationalists (individuals who strive to think clearly)  and often empiricists (individuals who base their beliefs on verifiable evidence) .
Rather than classifying all actions as either moral or immoral, the moral value of actions exists on a continuum from the extremely immoral to mostly neutral to extremely moral. And we should view actions that “should be illegal” and “should be legal” as a closely related but not identical concept. Sometimes making an immoral action illegal or punishing it harshly can create more harm than good, for example. We give individuals a lot of freedom in choosing their actions, because they may be better judges of what a situation calls for then any rule that can be made to apply to everyone. But everyone can always to do better.
As you can see, a big part of being ethical is determining the impact of ones actions. Most moral dilemmas actually don't come down to philosophical differences, so much as differences in belief about what the consequences of actions are. They are essentially political and scientific questions. Atheists tend to support liberal democratic ideals , beginning with the enlightenment  and progressing ever since . The median atheist  is probably slightly left of Obama . We atheists are generally more likely than non-atheists to support secularism, scientific research, free speech, education, humanitarianism, environmentalism, anti-war, anti-racism, anti-sexism, democracy, equitable distribution of wealth, and sexual freedom. In many cases, atheists views track closely with those of scientists .
So as you can see, the moral values of atheists are easy to state (wellbeing of living things), but determining the correct moral action can involve considerable psychological and political knowledge.
Maybe the way that I work through some of these specific examples will give you an indication of how atheists think about practical ethical dilemmas.
Let's start with vigilante justice . Suppose, hypothetically speaking, someone murders my sister, and I am considering murdering that person in retaliation. Generally this is wrong. The first way it can be wrong is that it doesn't bring my sister back, and killing her murder may not prevent further murder. If I kill her murderer, then someone who cares about him may in turn murder me and the blood feud  may spiral out of control. I were to claim my willingness to kill in retaliation for my sister is the sort of thing that would deter  someone from killing her in the first place, it would also incentivize that person to kill us both to prevent the retaliation. When violence occurs between two groups without mediation, they become enemies and the feud may escalate until one side is completely ruined and the other side is partially ruined. The best way to prevent this disastrous outcome is intervention by a third party, more powerful and trustworthy than either side. In this case it is the State and its monopoly on violence  which provides the intervention. If the state prosecutes my (hypothetical) sister's murder, than any retaliation against the state would not succeed and the murder's family will want to rely on the protection from the state. We know now from history that this strategy has been very successful and reducing violence in general (see ) and this presents another reason to turn to the state for the prosecution of crimes . Only in the complete collapse of society would I consider retaliation, even then I may prefer nonviolent restitution  if I believe it can succeed. In the case of a corrupt or ineffective criminal justice system, or in the case that I am already in a struggle for me and my own families right to survive, I might also consider waging a war - but even then I should aim for a better, fairer, more peaceful political order not vengeance.
I consider your scenario of “robbing rich guys who made wealth through illegal means immoral” in the same spirit. If I can rely on the state for prosecution I should do that instead. Even if I fail to get the state to prosecute in this specific case, if I can rely on its for most circumstance like this than I should seek only to reform the state for better results and not personally administer justice. Only if these rich guys completely control the state and are preventing it from prosecuting cases fairly would I be justified in robbing them - but even then their greater crime is controlling the state unfairly not their illegal wealth. If I were to rob them, then I would then also have a responsibility to either return the wealth to its original owners or use the wealth to create a positive and fair change in society. In each case, my action has ramifications beyond the immediate and I have to consider those ramifications the best I can.
Let's talk about prostitution next. I certainly think it should be legal. I find that most people who believe prostitution should be illegal believe that all prostitutes are coerced into it. In fact, there is no contradiction between legalizing prostitution and prosecuting also individuals who make money off of sexual slaves. In addition, legal prostitution gives prostitutes greater resources to protect themselves from assault or theft by customers or managers. Prostitution should also occur exclusively between adults and with proper precaution against the transmission of STDs. The Netherlands might be a good model for legalized prostitution, although they are still improving their system . Some argue prostitution is too awful of a job to be legal, but anyone who has the prostitute's interest at heart should be for legalizing it. If a prostitute voluntarily takes on the job, then it must be better for her than living in poverty. If that still seems unacceptably cruel, than we should fix the problem by treating our poor women better (economically speaking), not by criminalizing their strategy for making a living in adverse circumstances.
Other people worry about ill effects on family life, by legalizing prostitution. These people charge that men will never fall in love and start a family if prostitution is an available option for sex. These fears have not been born out in Dutch society, which is healthy and stable. In fact, men desire more than just sex, and men do not have to be bribed with sex in order to become good fathers and husbands. A healthy relationship should be formed on trust, not by controlling the other person. If monogamy is important to a marriage, a man should be willing and able to abide by it voluntarily. In addition, non-monogamous marriages can also be healthy marriages (more on that later). Maybe these things are not the sort of outcomes that could be predicted by a traditional society, but they have been shown to work in a modern society . The last thing I would say about prostitution is that it is likely not the most ethical expenditure of money. Anyone who has money to spend on a prostitute probably has money to donate to a more charitable cause, and they should do so instead. But that doesn't mean it should be illegal for them to pay for a prostitute. Here we arrive at something of a paradox - people, especially rich people, do not give as much money to charitable or humanitarian causes as they should . Prostitution could be better than whatever they were going to spend their money on, or it might not be. In general, though we count on individuals to make decisions like that for themselves, so I come down on the side of legalization.
Next we have pornography. Obviously, for pornography produced by filming individuals engaged in sexual acts, prostitution is involved and the same rules should apply to the production of pornography as in prostitution. No one should use pornography produced under unethical circumstances, such as non-consensual sex, sex filmed or published without consent, sexualization of children, or the use of actual (i.e. not pretend) violence. Other than that, I think it is permissible to watch and also to pay for pornography. Some people worry that pornography pollutes the mind, and encourages immoral behavior. Most of the negative effects have turned out to be pure superstition, but one potential negative effect merits further discussion. A varieties of psychological studies, like these [36-39], have shown that when men view sexual material they are primed  to not only sexualize women but approve of sexual assault and ignore a woman's perspective in moral scenarios. It is unknown whether these effects are generated because of negative cultural stereotypes about promiscuous women  or are more intrinsic to human psychology than that. Sexual Objectification  is the term used to describe the theory that the effects shown in these psychological studies contribute to sexism, the unfair and cruel ways women are treated, that we see in every society. The problem with this interpretation, however, is that no studies (to my knowledge) have shown these priming effects to be anything more than temporary. However those studies are very difficult to do effectively so we cannot rule out the possibility that pornography contributes substantially to sexism. We can only say the overall rise in Western consumption of pornography (from the Internet) has actually correlated with a decrease in sexual assault and sexist attitudes against women . But this data is also difficult to draw conclusions from, because cannot say for sure whether it was the pornography that caused this effect . To me, the best course of action seems to be for society to be very permissive of pornography while also being very proactive about discouraging sexism and protecting women's rights (most Western atheists come to this same conclusion).
Next onto “sex with multiple partners without marriage” is perfectly fine, given several caveats. The first caveat is that an individual has a responsibility to use protection to avoid the spread of STDs and unplanned pregnancies. The next caveat is that an individuals has a responsibility to be honest with all their partners about their sexual activities. Deceiving individuals that one is romantically or sexually involved is likely to cause misery. To put it another way, honesty will better guarantee that one's actions do not harm others . Even if we declare jealousy to be an irrational and useless emotion, it is still a real emotion that can cause real misery. One should have the happiness and well-being of each partner at heart (although there is no obligation to treat them all equally). Most atheists and many Westerners believe that the experience of having romantic and sexual relationships with multiple people (either in sequence or simultaneously) provides the best information for selecting a good match for successful marriage. Most atheists expect this sort of dating period to be temporary, before transitioning into a marriage that involves raising children together.
Almost all atheists believe in marriage, however, recently the community has to started to question the value of monogamous marriage. It seems clear that everyone should have enduring close relationships such as a marriage, close friends, or family. It also seems clear that it is also important to raise children in a stable household . But after the invention of contraception, its not clear why sexual and romantic fidelity needs to be a requirement for meaningful romantic relationships and responsible child-raising. Most Westerners associate “polygamy” with sexist traditions, in which several females are subordinate in marriage to one male, so liberal US atheists have embraced the term “polyamory”  instead to describe non-monogamous permanent relationships with any combination of equal partnerships . Unlike the other moral issues I've described so far, polyamory is something of a social experiment. We simply don't know the full ramifications of permanent non-monogamous relationships in a modern minimally sexist society. Right now I think atheists are cautiously permissive. At the very least, it seems to require a great deal of trust, communication, and maturity and I don't know that this will be successful for most people. If I meet individuals who say they are happy and in a polyamorous relationship, I will be friendly and will not discourage them . I have a lot of trust in the power of individuals to find the situations that they work well in. And I do find polyamory a fascinating trend that I've been following closely. Here are some articles to read more about it .
And the last thing you asked about is incest. A typical case of incest involves child abuse by family members (usually a step-father). Even when incest is a relationship formed by adults, if the older adult helped raise the younger adult than there is a strong potential of it becoming an abusive relationship. Children born of an incestuous relationship have a higher probability of developing genetic disorders. Those are three examples of how it can be wrong and usually is wrong. But I don't think it is intrinsically wrong. We can consider man who was raised by an adopted family and who falls in love with a woman outside of that adopted family. Then they discover that they are actually siblings! In this case, however, the only harm their incest would cause is the possible birth defects of their children - otherwise they might as well be strangers. If they chose to adopt (or not have children) than they can ethically proceed with the relationship as though it were any other. To think about their biological relation may feel disgusting, but feelings of disgust are not always a reliable indicator of moral value. Atheism requires actions be evaluated in terms of actual harm, and in this scenario there is no harm caused by the incest.
Some people would be dissatisfied with my introduction to atheists morality. They would say I didn't sufficiently justify the golden rule and why morality focuses on human well-being. Whose to say that the most important thing in the universe isn't building the largest stone pyramid? This is the realm of metaethics, a discussion of what type of facts ethical facts are and how we know those ethical facts are right. Here atheists actually tend to disagree. Some say moral values are based directly on physical facts, some of them say moral values are real but an abstraction upon physical facts, some say moral values are imbued with meaning only by living things, some say moral value are nothing more than popular ideas, some of them say we can't know which type of thing they are, and some of them say it doesn't matter which type of thing they are. The practical consequence of these debates seem to be fairly limited, since atheists generally end up having the same moral values, life and happiness. that I describe in the section above. I share some of my thoughts of the subject in the last part of this AllExperts post . Some people use meta-ethical arguments, like Hume's , to argue that atheism cannot support a moral system but actually these arguments apply equally well (or more effectively) against religious moral systems, so its a moot point.
The one philosophical point that actually seems to matter is the relative balance between human life and human quality of life. This comes up, for instance, in making healthcare choices  or deciding the ideal population size . One empirical way to establish the relative balance is between life and quality of life is by using an economics technique known as revealed preference  and introducing a universal unit known as a QUALY . If an individual is willing to trade some of their time being alive (or risk being killed) in exchange for an increased quality of life, than it must mean (to that individual) that the increased quality of life has a greater (moral) value than the life lost (or risk of life lost). This is at best an estimate, however, since people are a bit irrational and this method only works perfectly on perfectly rational agents - people can be inconsistent with others, inconsistent with themselves, and they can be limited in knowledge about what they truly want.
 Any moral argument that could be made for helping oneself can also be made for helping others. In mathematical language, it has to respect the symmetry of the problem. Let moral value be a function of physical facts, from a physical perspective all individuals have the same claim to personhood, therefore the intrinsic moral value of all individuals is the same. A modern scientific understanding that there is no meaningful difference between people of different races, is required to extend personhood to all human individuals . On the other hand, the physical differences between humans and animals means that it is not required to value humans and animals equally but it does follow that if humans have moral value than some animals must also have moral value .
 Philosophically speaking, of course. In real life there is often a third option, to flee, which is usually the best and safest route in dangerous encounters: http://www.samharris.org/blog/item/the-truth-about-violence
 Again, Peter Singer comes to mind: https://www.youtube.com/watch?v=Diuv3XZQXyc
 Perhaps a certain cultivation of trust and willpower is a necessary requirement. Or maybe its just a program that works despite our intuitions. An atheist should always tries to establish their argument in verifiable facts.
 Here is a website that recommends effective charities: http://www.givewell.org/
 Dishonesty, in many ways resembles a kind of violence. dishonesty implies that you oppose someone's knowledge and power, whereas honesty implies you intend to work with someone and that you share in their fortunes. If one is serious about treating others as you would want to be treated, honesty seems like a basic requirement. You would want to be informed when your actions could harm another person. Circumstances in which harm is the intention (some harm is tolerated for a greater good), such as self-defense or justified war, also warrant dishonesty.
 Raising children is of great moral value because it helps ensure the success of subsequent generations. The children can either be the biological children or the adopted children of the parents, both are equally valid ways to raise a child. Also, the burden of child-raising may not need to be distributed equally - some individuals might never raise children and that's okay if they are still spending their time in a way that contributes to society. Also stable households are happy households with good parenting - modern psychology has revealed it need not necessarily be a biological mother and a biological father.
 One man matched to several women, One woman matched to several men, two women and one man each matched to each other, two men each of which matched to each of two women, a man matched to a woman each of which is allowed to have less important relationships with others, etc, etc.
 One interesting way to think of polyamory is as an orientation, similar to sexual orientation. Some people are mono, meaning they'll be happiest in a monogamous relationship, some people are poly, meaning they'll be happiest in a polyamorous relationship, and a lot of people in between, meaning both types of relationships pose challenges.