The first thing I would look at before you decide which category of digital camera to buy, is what kind of photography are you planning on shooting?
The second thing I would focus on is whether or not your going to need different types of lenses, for instance, if your looking to to close-up shots that are full of detail, your going to need a camera that has a superb macro lens. In relation to this, dslr cameras are going to give you more flexibility with your lens variation and they will give you larger sensors which often lead to better picture quality when dealing with particular types of photography (especially when your blowing up or expanding the frame), you will see less pixels being blurred.
Third, I would look at what your budget is. There is often a considerable different in prices between dslr and compact point and shoot and or ultra zoom lens. The flip side of the coin is that if your simply looking for a convenient travel camera that is light weight and occasionally would like to take a couple of portrait, landscape and or slower ISO required photos, your options are abundant. You will also be dealing with a lower price bracket of camera options in most cases versus the dslr category.
There are many other variables involved, but to get you started, check out a couple of cameras in each category and use some reliable resources.
A few cameras in each category that I would take a look at would be (depends on price obviously, but these are highly rated):
Compact Point & Shoot:
Compact Interchangeable Lens
Panasonic DMC GH2
Panasonic DMX GF1
Olympus Pen E-PL2
Ultra Zoom Lens
Panasonic DMC FZ100
Nikon CoolPix P100
Pentax K-x or K-r
Nikon 3100 or 5100
As I mentioned earlier, it really depends on what type of photography your looking to shoot and what features are the most important. Anyways, good luck and enjoy shooting lots of fabulous photos.