Elo boosting, also known as MMR (Match Making Rating) boosting, is a service in which a high ELO player will increase someone else's in game rank by "boosting" them. This means the player (booster) gives the booster access to their account in order to fulfill the service. Once the booster has reached a particular league (Platinum, Diamond etc.) the boost is complete, and the player gets their account back. The owner of the account can then play in their new division or decide to take it further and start another boost.
During the boost, the account owner won't be able to log in or play any League of Legends (unless they have a separate smurf account!). This means they’ll either have to find something else to do to pass the time, or wait patiently.
This might all sound pretty straight forward, but why would you let someone boost your League of Legends account in the first place?