Arngeir is one of the 4 Greybeards in Skyrim. He is the unofficial leader of the Greybeards. He is the only one that will speak with the Dragonborn or any other Outsider of the order.
If the Dragonborn kills Paathurnax, Arngeir will refuse to talk to the Dragonborn anymore and remove them from the order.
Find the Word of Power - Radiant Quest