A friend of mine asked me the same question a few days ago
I don't know how new you are to photography, so I'm going to assume that you are very new. Get a SLR only if you're serious about going into it as a hobby. By serious, I mean wanting to go out just to shoot at least once or twice a month. I'm sure you know like any hobby, it takes drive and discipline to bear fruit.
If you're not sure how serious you are yet, get a compact, make the effort to go out for shoots, read up some fundamentals or take a course, and see if this hobby is really something you'd like to sustain. You'll also need to think about how you plan to use your photos? Print them out? Just upload to a web album?
Digital compacts make for fun and easy point and shoot sessions, and through their use you'll find out what it is you enjoy shooting the most. Then when you have a better idea of your personal shooting style, where you want to go with your photography, and are ready to go beyond the compact, go ahead and explore other cameras.
Another option is if you have friends with SLRs, ask to borrow theirs for a first-hand feel.
My personal preference for compacts are the Nikon Coolpix range. The Panasonic Lumix LX1 or LX2 are also nice, light cameras that are good for street and travel photography.