What is the in-universe reason for the name "Fairy Tail"?
From Makarov's history based on Wikia:


  The guild was named 'Fairy Tail' as the eternal mystery behind whether fairies had tails or not, or if they even existed, was like an eternal adventure.


So I guess the guild's (Fairy Tail's) purpose is to have a great adventure as if they are solving the mysteries about fairies.