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, but has hotshoe to attach an external flash. These cameras are slightly smaller than an SLR camera.
Advanced Semi-Compact - No interchangeable lens, but has hotshoe. These cameras are slightly larger than a typical compact camera, but smaller than a bridge camera. The two models listed below have more controls outside of the camera (compared to scrolling through menus).
Compact - No interchangeable lens, and no hotshoe. Compact cameras have small sensors; this translates into poor image quality. They are not useful for serious photography. But if you do purchase one, then at least get 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 do not.
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.