1. Open the League of Legends client
2. Go to options (cog wheel icon by minimize)
3. Uncross/untick/leave blank the box by "Allow Peer to Peer Transfer".
How it works:
The game installs a program called Pandoo Media Booster which uses P2P to help you update and load the game. It works in a very similar way to torrents where you're expected to upload any "pieces" you've fully completed to other players and receive the pieces you need in return to finish your client update. There is a possibility that your client will simply keep uploading after you've logged on and while you're playing a match (known as "seeding").
This method helps Riot by reducing stress and costs on their servers. Uploads use a greater portion of your internet bandwidth and causes lag not only in LoL but other internet applications.
By turning this off, it ensures you're only downloading from Riot's servers and there is no P2P being used which should turn off uploading.
1. Who should turn off P2P?
People with slow and/or laggy internet connections, although anyone can do this.
2. Why is it on by default?
It benefits Riot by not letting your client view them as the sole supplier of files that you need by providing multiple sources in the form of other players with the feature enabled.
3. Why do updates never work when servers go down and I have P2P on?
The same servers allow you to login. You can try updating on a different region so you're almost ready for when your preferred region is back online. There are region specific files that you must wait to download.
(Jokes beyond this point)