When people talk about "Full Size" or "Full frame" they are talking about using a camera system originally designed for 24x36mm film format with a digital sensor that is also 24x36mm format. When they say "Crop" format this means that the sensor format is smaller than 24x36, so the sensor is only taking in a part of the image (which is actually a circle - even a 24x36 film frame is only taking part of it).

The view given by a lens depends on it's focal length, expressed in millimetres. The more the number of millimeters, the more like a telescope the lens is. People have become accustomed to talking about lenses in terms of millimetres which correspond to a certain view when used on a piece of 24x36 film:

28mm = wide-angle
50mm = normal
105mm = mild telephoto
300mm = long telephoto

When you use a smaller sensor then you are effectively cropping off the edges of the image. The result is that you get an image which is more telephoto. In order to know what is the equivalent view in 24x36 terms with a given lens on a given sensor you multiply by the correction factor. Most DSLR's use the APS-C sensor size which has a correction factor of 1.5. So a 28mm becomes the equivalent of a 28x1.5=42mm which makes it a sort of "normal" lens rather than a "wide-angle".

The correction factor has no effect on picture-taking. You just put the lens on the camera, look through the viewfinder and fire away.

The advantages of using a "crop" sensor:

- much MUCH cheaper to produce than a larger sensor
- only uses the central part of the image made by the lens so no problems with loss of quality and light fall-off at the edges
- allows smaller and lighter cameras and lenses

The advantages of using a "full frame" sensor:

- the sensor is physically bigger so it can capture more light. This means a better compromise between image quality and sensitivity
- people can carry on using their lenses just the same as they did with film

Personally I think the whole thing is silly. It's the results that count.