Back in the days of film photography, there was only one sensor size in 35mm photography: 24mm x 36mm. So, when people refer to “full frame” cameras in digital photography, they're discussing 24×36 sensor size.
Unfortunately, full frame cameras also tend to come with a hefty price tag. The cheapest full frame Canon camera, for instance, is a few thousand dollars. Professional photographers use full frame cameras the most, because they need the extra features. The alternatives are “cropped frame” cameras, or “crop sensor” cameras. These have a much cheaper price tag, which makes them much more attractive to those starting out with DSLRs.
A cropped frame or sensor is similar to taking the middle of the image and discarding the outside edges. You’re essentially left with a slightly thinner image than normal—similar in shape to the short-lived APS film format. In fact, Canon, Pentax, and Sony usually refer to their cropped sensors as “APS-C” cameras. Just to confuse matters, though, Nikon does things differently. Its full frame cameras go under the moniker of “FX,” while its cropped frame cameras are known as “DX.” Finally, Olympus and Panasonic/Leica use a slightly different cropped format known as the Four Thirds system.