This is a new phrase that has recently become popular through social media websites. A "friend request" is a message which someone sends when they want to connect with you on a social media site.
If you "accept" a friend request, you and the person who sent the request will be marked as "friends" on that site.
The idea of "friend requests" became popular mostly through Facebook.
Some other social media sites connect people differently, through "following". Twitter works this way. When someone "follows" you, you don't have to accept the request or "follow" them back.