There are two basic types of cameras for personal use on the market: Point-and-Shoot and Single Lense Reflex, more commonly called SLR.
By far the most common are thos


SLR cameras are those which provide the most flexibility in taking a photograph. They allow the photographer to change how the camera is taking the photo, as well as the format it is saved. SLRs also provides interchangeable lenses for different effects. These cameras provide the greatest range in what the photographer can do as well as better image quality, but they are therefore more complex. This creates a need to learn how to use the camera before actually taking a photograph. Another drawback to these types of cameras is that they are much more expensive than Point-and-Shoot cameras.
Here is a quick comparison:
Point-and-Shoot - Small, cheaper, less features
SLR - Larger, more expensive, greater control of the image