Expanded Definition

In photography, aspect ratio refers to the ratio of the width of an image to its height. Most digital cameras have an aspect ratio of 3:2, which is the same ratio as a standard frame of 35mm film. Some cameras have a 4:3 aspect ratio, which is the same as a traditional TV screen. And some newer cameras have a 16:9 aspect ratio, which is the same as most HDTVs.
So why do different cameras have different aspect ratios? It all has to do with the size of the sensor inside the camera. A camera with a smaller sensor will have a smaller field of view, so it will need a wider aspect ratio to capture the same amount of scene. For example, a camera with a 1/2.3" sensor will have a field of view that's about 6 degrees wide and 4 degrees tall. But if we take an image with that camera and crop it down to 3:2, we'll end up with an image that's only 4 degrees wide and 3 degrees tall. So we'd need a wider aspect ratio to get the same field of view.
On the other hand, a camera with a larger sensor will have a wider field of view, so it can get away with a narrower aspect ratio. For example, a DSLR with an APS-C sensor will have a field of view that's about 18 degrees wide and 12 degrees tall. So if we take an image with that camera and crop it down to 3:2, we'll still end up with an image that's about 12 degrees wide and 8 degrees tall. That's plenty big enough for most purposes.
In general, then, you'll want to choose an aspect ratio that matches the way you want to use your images. If you're going to be printing them out or viewing them on a traditional TV screen, then 3:2 is probably your best bet. If you're going to be viewing them on an HDTV or sharing them online, then 16:9 might be better. And if you're not sure what you're going to be doing with your images, then it's probably safest to just go with 3:2. That way you'll always have the option of cropping later if you need to.