According to the major league rulebook, Section 10.15b "A catcher shall be charged with a passed ball when he fails to hold or to control a legally pitched ball which should have been held or controlled with ordinary effort, thereby permitting a runner or runners to advance."
In other words, a passed ball is when a catcher lets a ball go past him which he should've caught, allowing a baserunner (or the batter on strike three) to advance as a result.
Passed balls are somewhat rare and while stats are kept, it's one of the more obscure baseball statistics.
The most famous passed ball happened in 1941 when Mickey Owen of the Brooklyn Dodgers dropped a 3rd strike in the 9th inning of Game 4 of the World Series against the New York Yankees. If Owen had held on, the Dodgers would've won and tied the Series 2-2. He did not, the batter reached first base, the Yankees ended up scoring 4 runs in the inning, winning the game, and winning the World Series 4 games to one.