For serious photography hobbyists, there are several items to consider when purchasing a camera.
A hotshoe (used to attach an external flash to the camera). Compact cameras do not possess a hotshoe.
How many megapixels does the sensor contain? Generally, more megapixels = poorer image quality. If I were comparing two virtually identical cameras, but one had 16 megapixels, and the other had 20 megapixels, I would choose the 16 megapixel camera.
The "size" of the sensor; generally, smaller sensor cameras will produce poorer image quality.
The ability to switch lens on the camera body is desirable. But if you do buy a camera with a permanent lens, then be sure that the zoom range is acceptable for the type of photograpy that you wish to do.
Manual exposure controls. The ability to manually control the exposure is quite useful when shooting under certain circumstances. Such cameras will have aperture & shutter priority, as well as a manual exposure mode. The specific models listed in the "web links" below, do have manual exposure controls for aperture & shutter priority
SLR - allows for interchangeable lens, and has a hotshoe on top of the camera to attach an optional external flash. Below are two entry-level SLR camera models.
SLR-like (aka "bridge" camera) - has a permanent zoom lens. No interchangeable lens is possible. These cameras are slightly smaller than an SLR camera.
Compact - No interchangeable lens, and no hotshoe. Compact cameras have small sensors; this translates into poor image quality when shooting indoors. They are somewhat limited for serious photography, unless you at least purchase one that has aperture and shutter priority. Listed below are a few examples of compact cameras that do feature aperture & shutter priority; the vast majority of compact cameras do not have aperture & shutter priority.
Note: If you already have some camera models in mind, you can search out their features at dpReview.com (Be sure to click the link that says "Camera COMPARE." You can then select the camera models you're interested in and compare their features; When you eventually get to the "comparison table," you'll see that Aperture & shutter priority will be listed under the "Photography Features" section of the table.