Many people new to photography will hear the terms "Full Frame" or FX and APS-C or DX format. But what does this really mean ?
Well the technical definition involves the size of the sensor in the camera. The sensor is the digital equivalent of film. i.e. it's the light sensitive part of the camera.
On an old film SLR, the film size was what was called 135 format. It measured 36mm x 24mm. In the digital SLR world, this format has become known as "Full Frame" or for Nikon people, FX format. The sensor is the same size as the old 135 format film.
In the mid to late '90s a new film format was introduced called APS. One version of that standard was APS-C. The film size was around 1.5x smaller than 135 format. Similarly, in the digital world, APS-C sensors are 1.5x or 1.6x smaller than their Full Frame counterparts.
(Typically Canon sensors are 1.6x smaller, while Nikon, Pentax, etc use 1.5x smaller sensors - Nikon calls this DX format).
Now what does this mean in the real world ?
Well the field of view that an APS-C camera captures is 1.6x smaller than a Full Frame camera for the same focal length. i.e. the outside portion of the frame is chopped off, or cropped on an APS-C camera when compared to a Full-Frame camera.
Here's an example using a Full Frame camera (Canon 5D MkII) and an APS-C camera (Canon 30D). The same focal length, distance to subject and exposure was used on both images.
Notice that although the field of view changes, the depth of field stays the same (pay careful attention to the amount of the wire that is in focus), but that on the APS-C image there's an apparent increase in depth of field as more of the frame appears to be in focus. This is a consequence of the cropped nature of APS-C cameras.