Since I take a lot of pictures, I'm often asked for advice by people looking to buy a camera. Here's the general advice that I give them.
As with every equipment purchase, the item you settle on is the result of a number of trade-offs. You can therefore give up on getting the "perfect" camera. There will always be things that you won't be able to photograph, things that will be difficult to photograph and, hopefully, the vast bulk of things that are easy. The trick is to have the "bulk" overlap the things you are interested in photographing to as large an extent as is possible.
Instead of diving into the technology - you can do that elsewhere - what I hope to provide you with is a way to understand how to evaluate the technical features of a camera in terms of relevance for you.
First, in my mind, is the price. Money spent on a camera is money not spent on going where the interesting things are. If you ruin yourself on a Leica or Hasselblad, you won't be able to go to interesting places and take interesting pictures.
Second is the mobility. If you don't have the camera with you, you're not getting any photos. In general, people tend to overestimate their willingness to haul things around. It helps to think about a point-and-shoot as a big second cellphone, and a DSLR as a small backpack. Would you carry those things around everywhere you go?
Third comes the feature set, and here I can reassure you: In a given size and price class, all cameras are pretty much the same. Therefore, we can start small and cheap, and then add features until our wallets or backs hurt.
How to evaluate a feature then? Here's my method. I think about:
The number of photos that were ruined because I didn't have the feature.
The amount of time I could have saved if my current camera had the feature.
Salespeople will try to make you think about what you want to do, instead of what you actually do. Have you ever been in a situation similar to this: The salesperson asks you "do you want to do any low-light photography?", and at the same time, shows you some incredible low-light shots. Perfect framing, perfect composition, perfect subject. You haven't shot a single low-light picture in your whole life, but immediately you think "oh man, I want to take pictures like that so bad I'm going to die if I don't", and go "yes, sell me something that can do that!". Once you've been asked about nature photography, sports, landscape and macro - all of which rack up the price - you walk out of the camera store thinking that putting a cool $40,000 on a camera and ten times that on lenses is not just a good idea, it is vital to your survival. If you instead think about the number of shots you've missed due to poor low-light performance, you would have known that it is exceedingly rare and therefore not worth much money to you.
On the other hand, if you really do a lot of low-light shots, then you'll remember every single shot that turned out bad because of the small sensor or slow lens you used. Plus, you'll know exactly what you need - faster lens or different camera - to fix it.
To illustrate the second point, I'll use myself as an example. I used to have a Casio EX-Z850 point and shoot. It was great, but the lens was a 37mm equivalent and I just love wide-angle photos. At first I used Hugin, a panoramic stiching program, to make wide-angle shots from a set of component images. The results were good, but every photo took about half an hour to an hour to process, and I never really knew if it would turn out well in the end. When the camera broke, I thought I'd take the plunge and get a DSLR. I knew I could get a Nikon D40 for about $400 on sale and a Sigma 10-20mm lens for about $600. Together, they'd cut the per-shot time from 45 minutes to, well, nothing. At the same time, $1000 was worth it in my mind, and the D40 small enough that I knew I'd be able to carry it around. So I went for it, and ended up really happy.
The point here is that I knew exactly which feature I wanted: The ability to attach a 10mm lens. I knew what I was willing to pay, and I knew I was willing to lug around a D40.
So to summarize: Unless you really, really know otherwise, assume that you'll do the same kind of photography you do now. Then focus on how the new camera will make that kind of work easier.
Update 2010-12-09: I have written a couple of follow-ups about this. They're all filed with the label Buying a Camera.