Other camera types that I can think of, in terms of digital, would be:
Cell Phone - Destined, I think, to ultimately replace, or at least supplant, the point and shoot. They currently represent what is the most likely camera for a person to be carrying and that is what gives them advantage: the camera you have with you is infinitely better than the one that you do not. :) At any rate, they're getting better.
Bridge - The basic body structure of an SLR with a built-in lens and P&S features. The name, as implied, is about bridging the point and shoot to the SLR. In my opinion, and it is strictly opinion, you end up with the disadvantages of both (bulk of an SLR, image quality of a P&S) without their advantages (lens interchange of an SLR, portability of a P&S).
Medium Format - Big, powerful, and a price to match. The cheapest of these start at $10,000 and it skyrockets from there. The advantage? Image quality. The medium format camera represents the best of image quality, but they're bulky, have slow frame rates, and more restricted ISO ranges. So, there are trade-offs there, even for the price.
Mirrorless - It can be micro four-thirds or the more radical Nikon V1 and Pentax Q, but these are interchangeable lens cameras without a mirror. They're often equipped with electronic viewfinders. Their advantage over P&S is usually in lens options and image quality, their advantage over the SLR is in size.
Rangefinder - Not an SLR as it doesn't have the moving mirror. Leica is the king of this market. Big advantage is smaller body over an SLR, but they don't lend themselves well to zoom lenses because of the rangefinder.
However, even in the two you really know about, the variation is enormous. Not all P&S cameras are made equal and for the SLR world you have:
Entry Level - Almost a point and shoot, but with interchangeable lenses. The Canon 600D/T3i or the Nikon 3100 are great examples of this.
Prosumer - Pro level features, consumer styling. The Canon 60D or the Nikon 5100 are examples.
High-end Prosumer - More pro features, such as weather sealing, faster frame rates, etc. The Canon 7D or Pentax K-5 are good examples.
Professional - Usually full-frame sensors the size of 35mm film. These are top of the line cameras such as the Canon 1DX or the Nikon D700.
Note that some of these cameras blend or bleed between tiers, but the feature sets on the cameras do tend to lend themselves to one of those 4 tiers.
Anyways, that's my synopsis. I'm sure others could expound upon this much more.