I have a fairly new monitor but a very old Dell desktop pc. The monitor has 2 ports, one VGA and one DVI (it came with a cable for each) but no HDMI port (I don't know why, it came out 2 years ago). Since my computer tower has just VGA, that's how it's connected. I plan to get a new Dell desktop(Inspiron model I3847-3850BK), but from the description I see for that and other Dells, the tower has just HDMI and VGA ports, no mention of DVI port...or is it called something else? Assuming it doesn't have one and I want to connect the monitor by using its DVI port, would I need an HDMI adapter or a VGA adapter? I emailed Dell and I don't think they really read my question, they said I would need a VGA adapter. But a computer store I asked said there would be no image with that and I would need an HDMI adapter. Would that be the way to go? I'm confused...thanks.
If the monitor has a VGA input, and the new computer has a VGA output, you can connect things that way via VGA cable. That would probably be the easiest as it's what you already have in place, so you could re-use the cabling. However if you want a digital connection, HDMI and DVI are essentially the same video signals, and you just need an HDMI to DVI adapter (and "they" (being cable makers) produce HDMI to DVI cables that are fairly cheap) to run between the two. The only catch is if the monitor requires a resolution higher than single-link HDMI/DVI can handle (roughly 1080p), in which case you would need to use VGA. The only potential benefit to going with a digital connection is HDCP, if the monitor supports it, but this would only generally be required if you're going to playback HD-DVD or Blu-ray on the PC.
As far as adapting VGA to DVI or similar, it requires analog to digital conversion (VGA is analog, DVI/HDMI are TMDS digital) - adapters do exist, but they're much more complex devices than an HDMI<->DVI cable, and they're directional (meaning they only go from one to the other, not either way). I wouldn't suggest that route unless you have absolutely no other way to connect things, which does not appear to be the case here.