I've talked with a bunch of people recently who were looking to buy a digital camera so I wanted to give my plug for one of the best ones around right now. Basically, if you're an average user like most people, then as long as you have a camera that is 2.0 megapixels or larger (which is almost any new camera you buy these days ... and megapixels are basically the number of dots in the picture ... the more dots, the finer quality photo you'll have), the picture quality should be fine . If you want to blow up pictures to larger sizes, then you want more megapixels. And in terms of quality, shutter speed (normally I like Sony products, but Sony is a bit slow in this area), functionality, size, etc. my recommendation would be to get a Canon. Most reviews I have read seem to agree that Canon is one of the best brands out there right now. And the best Canon series (for the average user) would be the Canon IXUS/Powershot Series.
Several people in our EM (Eric [who also did a ton of research before buying his too], Bomi, Stang [who is a photo-maniac!]) and I all have this series and we all love it! No problems, just clear pictures with a lot of fun options (like a rapid shot function where you can take like a billion pictures every second ... makes for some fun film making and great for sports photos). Hopefully Eric (EBae) can take it from here and rave even more about our camera :p But this is definately my pick for most people.