What is the difference between Full frame and cropped sensor?

We are talking here about digital cameras, and specifically digital SLR (single lens reflex) cameras, which have interchangeable lenses. A full frame sensor is the same size as a 35mm film frame. This means that if you put a given lens on a film camera (let's say a 50mm fixed focal length) and then the same lens on a digital SLR with a full frame sensor, both cameras will capture the same scene.

A cropped sensor, as the name suggests, is smaller than a 35mm frame, which means that the same lens is effectively zoomed in slightly, generally by a factor of about 1.6:1 (this ratio is called the crop factor). This means a 50mm lens on a cropped sensor camera gives approximately the same field of view as an 80mm lens on a 35mm or full frame sensor camera. This is because the smaller sensor only captures the light from the centre of the image.

There are pros and cons to each. One major advantage to cropped sensors is they are generally much cheaper to produce, which is why you typically only see full frame sensors on high end cameras (taking Canon as an example, only the 5D and 1D range have full frame sensors). They also allow the use of potentially lighter lenses, thanks to the crop factor effectively increasing the focal length of any lens and optical adjustments that can be made to the lens design thanks to the need to focus onto a smaller area. As far as I am aware (and certainly in the Canon range, which I use) a cropped sensor camera can use any lens designed for its 35mm or full frame counterparts (albeit with the crop factor), whereas lenses designed for cropped sensor cameras (Canon's EF-S range being a case in point) cannot properly be used on full frame or 35mm cameras.

On the other hand, full frame sensors are often more sensitive, producing sharper images with less noise, particularly in lower light. The fact that they also allow a photographer to migrate from 35mm to digital without affecting the focal lengths of existing lenses may also be a benefit to some.