Clans are a way for players in FAF to form a bond as a group, may it be small or large. Any player can create or join a clan by visiting the FAF clan site. You invite other players by generating an invite link and having the player click on that link.
Clans have no effect on the game besides adding a clantag in front of a players name in the chat, the lobby and the game itself.
Aside from that, players who are part of a clan have their own clan chat room.
Since version 0.10.108 of the lobby, clan tags are displayed in front of player names.
Forged Alliance Foverer has its own Clan management https://clans.faforever.com/clans (it is still in beta but very