A bad beat is a hand in which a player whose hand was highly favoured is beaten in an improbable turn of events. The boundary between a regular beat and a bad one varies from player to player. It's is generally thought that a winning hand whose chances of winning were around 5-10% is a bad beat, or if a player wins by hitting two perfect cards.
Here is an example of a bad beat in Texas Hold'em:
Player A Player B Flop Turn River
Player A has a straight on the flop, B just a pair of nines. But on the river, B wins with a full house. The chances that B would win given the flop are around 7.8%.