Achievements are a future content item of League of Legends, however there has not been any mention to them for quite some time. The "Achievements" tab could be seen in a player's profile in-game, though it could not be selected.
It was removed as part of the 3.7 update in which a new "Item Sets" tab took its place.
According to the tab in which you chose your summoner icon, players will be able to unlock more summoner icons by completing achievements.
There has been no further information on implementing the achievements or whether or not it would be returning.