Jim
wlc1 is correct, a coil spring fork is exactly what is says, as normal coil spring that absorbs the shock under compression inside the fork, an air spring is a sealed chamber inside the fork that is filled with air at a set pressure, the air then compresses to act as a shock absorber, most forks allow you to change the pressure of the spring for coil spring forks, or change the air pressure for air springs.
Why use one or the other:
Coil spring, cheap for lower end forks and also as they are very robust and able to take much bigger shocks than air springs they are used for the big hitting down hill and free ride bikes.
Air Springs are used as they are lighter.
IHTH
Giles