Full Frame: A sensor equal to size of a 35mm film (24mm x 36mm)
Crop Sensor: A sensor smaller than 24mm x 36mm that has a 'crop' factor (Nikon's DX models have a crop factor of 1.5, Canon has two smaller sensors with a 1.3x and 1.6x factor)
For almost all applications, the Full Frame sensor has an advantage, mainly due to it's reduced noise, especially at high ISO. Full frame sensors are also better with wide shots due to know crop factor. Where does the full frame fall short, well in price, weight, size, and losing the extra reach that a crop sensor can give (A 200mm lens on a crop equals 300mm due to the factor, where on a full frame all you get is the 200mm).
So it seems to come down to what you are shooting and who your photographs are targeted for. For most wedding, architectural, landscape (actually would go medium or large format, but that is another discussion), or anyone who works in low light a full frame camera would be the best choice.
For the average consumer the price of a crop camera and lenses is the way to go. Also for sports/wildlife enthusiasts the extra reach of a crop sensor might be beneficial.
The one thing to keep in mind if you are making the jump from crop to full is that all your crop lenses may not work on a full frame sensor since the image circle is larger on the full frame.
Downtown Chicago from the River!