FPS, "Frames Per Second"

FPS, or Frames Per Second is the amount of frames being flashed on your screen per second to deliver moving images. 
Even though it's not everything,  FPS is very important when gaming, as a low FPS will leave you at a tiny disadvantage against someone else with a higher FPS.
Your monitor and GPU (Graphical Processing Unit, see Graphics and Video for more info) are the two main factors that determine your FPS. 

Every monitor has a refresh rate, or how fast it can flash frames. Most modern monitors have a refresh rate of 60Hz which is average for gaming. Monitors with a refresh rate of 120Hz is optimal, but not necessary for gaming (they're expensive too!).

A monitor with a refresh rate of 60Hz it limited to displaying games and video at 60 FPS, because that is how fast the monitor can flash images.

With a 120Hz monitor, you can run games and video up to 120 FPS. There is a huge difference in gameplay, which is why most hardcore gamers go for higher end gear.

If you're using fraps, a well known recording software, you can see how many FPS your game is running at while playing in real-time. Note: If you are running a game over 60 FPS on a 60Hz monitor, the game will be displayed at 60 FPS or even lower, even though fraps says higher. Same goes for 120Hz and higher/lower.

Your GPU is the key factor of any high FPS count. Things like smoke, explosions, collisions, and a lot of other graphical-intensive things will put a load on your GPU. Better GPUs can offer better performance under heavy load. See my page on Graphics and Video here