your first decision to make is portablility
do you want something that fits in a pocket ... or needs a big bag ... or something in the middle ish
if its got to go in a pocket so you can take it anywhere ... then look at compact cameras
if thats not an issue ... Digital SLR are fantastic .. but bulkier and higher price (but you can add to them over time)
then you need to look at what you want to take photo's of and how you want to do it
you get more control with a DSRL and the option to add better lenses that suit the picture types you want to take (longer zooms, wider angle lenses for landscapes etc, specific portrait lenses, ones that work well when its dark), however its perfectly possible to get some great shots out of a compact camera
you need to be aware of shutter lag (delay between pressing button and taking photo) is prob the biggest cause of bad photo's (only half the subject in it etc) .. well ... that and camera shake ...
DSLR has no shutter lag, compacts in general do ... so get the best one you can if you go for a compact, however the reverse is that DSLR often does not have antishake, or image stabilisation unless you buy a lens that has it, although this is less of an issue due to the size of the camera and how you hold it
hope thats not too confusing
Phil