Perhaps I'm getting ahead of myself. Maybe you don't have a digital camera yet. While this is a rather huge topic, I'll try and distill it to its most basic points.
Likely your first consideration. Most retail (i.e. not used) digital cameras will run somewhere between $100-$300 US/Canadian.
Much less than that and you'll likely run into problems. Very cheap cameras lose the pictures when you take the batteries out. They will be a very small and/or poor quality pictures even if they do save to a memory card. The least expensive won't have an LCD screen to review your snaps.
Sub $100 cameras aren't a bad starting camera for someone quite young - at least till their early teens. Expect it to be outgrown very soon, and more headaches when less money is spent.
Beware of stats. While they might be technically correct, they don't tell the quality of the components.
Below are 100% cropped shots of a new, cheap 9MP camera (top) and a old 6MP camera (used costs as much as the 9MP retail)
Megapixels really only measures the final size of the image. As you can see, a larger number here isn't automatically better. Any (quality) camera 6MP and up will print a passable 8x10.
While double the Megapixels is double the image size, it's still not as noticeable as one would expect.
In the image below, the 6MP is a 4:3 ratio Point and Shoot camera and the larger 12.2MP image represents a 4:6 ratio SLR camera.
If you are able to spend more than $300, there's also reason to be careful. If you are buying a point and shoot camera, you're in danger of being a victim of 'bleeding edge': Technology which is overpriced and not yet perfected. You might want to consider an SLR camera (more on that later).
In terms of dimensions, there is a huge spectrum: between small pocket-sized cameras to large SLR cameras.
While the practical benefits of a small camera are fairly obvious, there's a few things to keep in mind. First, you're paying certain premium for miniaturization. All other factors being equal, you'll pay more for the smaller camera. Second, in most cases, 'all other factors' aren't equal. The smaller the camera, the smaller the sensor size and the less zoom you'll likely get.
The image sensor is what actually digitizes the picture. While it's most often described in megapixels, it's also measured in a very real sense by physical dimensions. SLR cameras (and a few very specific Point and Shoot cameras) have a larger image sensor than most Point and Shoot cameras.
Even if they are both listed at the same Megapixels (which these below aren't, but close), the results are quite different. Canon SX110 (9MP top) vs Canon XSi (12.1 MP). Notice the 'flecks' in the top image, and how the bottom image resolves smaller details
Zoom is a slightly inexact measurement that represents how much a camera can bring a subject 'closer'. It's a relative measurement, since it compares the range to itself - the camera in question might start with a very wide angle, or start where a different camera/lens might end.
In any case, it's a matter of optics - a very small camera cannot magnify as much as a larger camera.