Spacewar!
MIT students — Steve Russell and others, who called themselves the Hingham Institute Study Group — wrote a spaceship duel for the newly arrived PDP-1. A star with gravity, ships drifting under inertia, finite fuel and torpedoes. It spread from lab to lab, copied tape to tape, and DEC used it to demonstrate its new machine. What was sold was a computer; what traveled along with it was a game. The ancestor of the arcade to come is here.
The myth, corrected
Spacewar! was the world's first video game
There were earlier ones. NIMROD in 1951 (a computer built to play Nim), OXO in 1952 (tic-tac-toe on EDSAC), and Tennis for Two in 1958. Which one counts as "first" shifts depending on the definition used. Spacewar!'s real significance lies elsewhere — it was the first game to spread beyond the lab that made it. When DEC shipped a new PDP-1, it kept this game loaded in memory, and engineers used it as the first demo they ran for customers. A game came bundled with the machine that was for sale.