The key questions: what sort of photos are you thinking of taking, and where are you planning on taking your camera?
If you take most of your photos in restaurants and house parties (i.e., indoors), you want good low-light performance and probably a small camera.
If you're going to be taking mainly landscape photos or want a good zoom, you may want to get a different sort of camera.
Also, are you an amateur, or one who wants to have manual controls?
Types of digital camera:
- pocket-size
- mega-zoom with fixed lens
- SLR
the SLR is more for your keen amateur who want more manual controls over everything.
The mega-zoom cameras offer much of what SLRs do, but without all of the control (you can't take the lenses out and replace them with others). However, they do generally have large zooms (10x optical or more -- only look at optical zoom, not digital, as vehelpfulguy says).
The pocket-sized cameras are much smaller but still take excellent photos. However, due to their size restrictions, SLR cameras will always be able to produce better images technically in many circumstances. For the average person who wants to record birthdays, holidays, etc., you will be more than happy with a compact.
Good cameras around now (I'll just list some compact cameras for now, since I'm thinking that they may suit you best):
Fuji F31
Casio S600, S770, Z850
Canon range (pretty much any one), specifically SD800
Nikon
Some Panasonic Lumix