"Carry" is a term that is typically used to refer to the primary damage dealer / killer on a team. So you will usually see a carry have the highest number of kills on that team, by the end of the game. Carrys may often start with low effectiveness at early game and rise to power through the mid / late game. The carry is usually a character that your team wants to feed, you want him to farm and get the best gear possible so he can own.
"Carry" is also a verb. When a player "carries" a team, this means he played the most significant role in helping the team win. Without him, the team would have had less of a chance, though this doesn't mean that the carry doesn't need teammates or that he does all the work, or that he is MVP of the Universe and has the biggest e-peen in the world.
"Carries" are usually squishier than tanks and not necessarily the most capable disablers. So they usually rely on their team to support, they ask their tanks to initiate into the fight and take the brunt of the enemy assault so that the carry can do their killing business. Carries also usually rely on their support heroes to disable enemy team members, but also rely on support for heals / shields / cleanses / resurrection / buffs. A buffed carry is a happy carry.
The carry role is less strict and less defined in LoL than in DotA, but it's still there.