A post on the League of Legends blog explains the details. You'll see a prompt after match-made games to click a green "thumbs up" icon on your own teammates or enemies. Everyone has a limited amount of Honor to give out, and you'll earn more by finishing more match-made games.
"For the average, responsible user, you will probably never run out of Honor," the blog notes. "However, players who spam or trade Honor will soon find themselves with nothing left in the bank to award. If you see players trading Honor, please report them using the 'Spamming' report! Getting punished by the Tribunal for trading Honor or toxic behavior will reset a player's Honor to zero across the board, however Honor can still be re-gained after the punishment has expired."
The blog also points out that Honor is not a currency and doesn't currently give any tangible rewards in the game. It does, however, give users an easy way of telling which players are helpful or respectful, which should make the whole game a bit more pleasant.