A magic number is a measure of how far a team is from being assured of finishing higher than another team. In hockey, they're typically expressed in points; in baseball, in games. You don't usually hear them used in other sports, as far as I know. For example, imagine Cornell has 27 points and Harvard has 20. Each team has 6 games left, for a total of 12 potential points. Therefore, the maximum points Harvard can finish the season with is 32. In order to assure a finish ahead of Harvard, Cornell then needs 33 points. The magic number is 33 - 27 = 6. Each point gained by Cornell, and each potential point "lost" by Harvard (1 for a tie, 2 for a loss) is subtracted from the magic number, and when the magic number reaches 0, Cornell has clinched a finish ahead of Harvard.
(This answer was submitted by Josh Herman '99)