What does the peer to peer option do?

Solution 1:

If you mean the option in the launcher, yeah, it's just to download content faster.

While playing the netcode of LoL (or any other game for what i know) doesn't work as a P2P network, there's a host and a group of clients. Usually you never exchange datas with other players, because if the netcode should rely on that, it would be easier to hack the game.

Solution 2:

If the P2P option is activated, you only share data about patch/update. No other informations are exchange between users.

If the P2P option is off, you'll have to download all the data from the server. This options is probably longer than the other one!