DX and FX are Nikon nomenclature. An FX camera has a sensor that is roughly the size of a 35mm piece of film. A DX camera has a sensor that is about half that size. You sometimes see DX cameras referred to as APS-C.
First, I'd like to dispel the myth that there is something special about "full frame" cameras. It's just an arbitrary size with no special properties. Sensor size impacts many things, but there is no single perfect sensor size. 35mm film wasn't chosen because it met some special criteria.
Bigger sensors have several advantages. They have more area for gathering light, so they tend to have less noise at any given ISO. The advantage is theoretically about 1 stop.
For optical reasons, you get shallower depth of field with a larger sensor. This is great if you want shallow depth of field (common in portrait, wildlife, and bird photography) but not good if you want more depth of field (landscape photographry).
Given a DX and an FX camera with the same resolution, the FX camera will have a lower pixel density. That means that it is mangifying the image from the lens less. That means that optical limitations and defects of the lens will be less apparent.
There are several downsides to larger sensors. They are bigger, which typically results in bigger cameras. They also require the use of bigger lenses. With Nikon cameras, I believe that you can shoot with DX lenses, but the camera crops the photo to match. With Canon, the EF-S (DX equivalent) lenses will not mount because of mirror spacing limitations.
When shooting FX lenses on a DX camera, you get some advantages. The center of a lens is usually the best optically and that's all that you are using. You also have fewer problems with vignetting (darkening of the edges and corners).
Smaller cameras magnify their images more. That means that the field of view will be narrower (more zoomed in) for any given focal length. A DX camera shooting through a 100mm lens will produce an image with the same field of view as an FX camera shooting through a 150mm lens. That's great if you want longer lenses (birds, wildlife) but it is bad if you want wider lenses (architecture, landscapes).
Outside of the Nikon/Canon world, there are additional choices. Olympus and Panasonic (I think) make an even smaller sensor than the DX cameras. Many companies make larger sensors than the FX sensors, but the costs for those systems are usually well into the tens of thousands of dollars.
It's hard to say what the future holds. For a while, it looked like DX was going to be the new standard. Today, it appears that Nikon and Canon are committed to both sizes. At the moment, it looks like the high end cameras and lenses are focused on FX sized sensors, but that could easily change. I know several people that could easily afford FX cameras but shoot 50Ds and D300s because they better match their needs.
