When shopping for a new digital camera, you want to look at four main things:
The resolution refers to the quality of your image, in terms of the number of pixels. The greater number of pixels, the better your resulting image will be.
This refers to the maximum opening of the lens, or how much light it can let in. When your camera has a large lens aperture, it can allow more light into the lens, which means you need less light in your photography environment to take a good quality photo.
Lens aperture is measured in f/numbers, such as f/2.0 or f/3.5. An aperture of f/2.0 means that the lens opening is half the focal length of the lens. Smaller numbers mean larger lens openings, so it's better to opt for a f/2.0 than an f/4.0
If you like to shoot a lot of indoor photos that are more natural looking, it is a good idea to have a fast (large aperture) lens. Photos taken without flash look more natural than those taken with it. It's also easier to take multiple shots without flash. It is easy to hurt people's eyes if you need to take 20 pictures in 10 minutes, all using a flash. You'll get better candids if you can take your photos without flash, simply because people won't even notice you taking the photos.
Cameras with a zoom feature will have multiple apertures listed.
The zoom range refers to the variable focal length of the lens, and the focal length determines the magnification of the lens. A short focal length is a wide-angle lens, great for taking in large vistas. A long focal length is a telephoto lens, allowing you to get a tight photo of a distant object or person. A zoom lens lets you get the best of both worlds within a single lens. If a camera has a 10x zoom, it means the longest focal length is 10 times the shortest.
When shopping for your camera, ignore the "digital zoom" you'll see on the ads. This is of no value to you, as it only refers to the ability to crop the photos from within the camera. You can do this with software on your computer after you remove the photos from the camera.
Even lenses with the same focal length and aperture can have varying quality. A poor lens is not as sharp, and it may not bring all colors to the same focus.
You won't be able to tell the lens quality just from reading the specifications on the box, so the best way to make sure you're getting a quality lens is to stick with trusted camera companies you know, and to read real user reviews on the specific models you are most interested in.