Spacewar!

According to Dr. Stewart's notes (found at Blackboard Academic Suite):

"Most historians of video games also hail Spacewar!, coded by students working at MIT in 1961-62, as an important milestone in the development of video games. Spacewar was developed on the experimental PDP-1 operating system which allowed multiple users to share the computer simultaneously. Spacewar could be played as two-player game involving warring spaceships. However, none of these early programs were consider to be important or worthy of further comment at the time."

Here you can read about the origin of Spacewar
Here you can play Spacewar online