A carry is a hero that can deal high amounts of damage but to compensate, has low Defense. Typically, a Fighter or Mage will take the role of carry but some heroes that aren't fighter or mage can carry. Carries are meant to have the most gold and kills at all points in the game and excel towards the end of the game at dealing very high damage to opponents.
Carries are key heroes in a team and are usually targeted first by the enemy team so that the carry can't deal his/her damage. Because of this, they need to be heavily protected by Guardian and Support in fights so as not to be taken out.
A carry can also refer to a ADC which stands for Attack Damage Carry or a APC which stands for Ability Power Carry. These terms are used in League of Legends but are also used in Heroes of Order and Chaos.