The Centauri run was a colloquial name for a Starfleet patrol assignment between the Sol system and the Alpha Centauri system.
Ships and personnel on the Centauri run surveyed Sector Zero for navigational hazards, as well as supplied the closest starbases to Earth. Newly-minuted ensigns just out of Starfleet Academy often drew this assignment.
Spock's first post-Academy assignment was to the Centauri run aboard USS Southampton, where he was billeted as assistant science officer. ("The Crash")