A carry is a term describing a champion or player that largely influences the outcome of the game by either dealing high amounts of damage or by providing decisive utility in form of crowd control or peeling. In its most primitive definition, a carry is a player that contributes majorly to the win of the game for your entire team, and is as such mainly used to describe a champion with immense damage, as crowd control and peeling are valued only by the more experienced players.
Classes often referred to as carries are marksmen, mages and assassins, or as in old terminology attack damage carries (ADC's) and ability power carries (APC's).