Master is the title given to the humans who participate in the Holy Grail War.
In principle you must be a magus to become a Master, but in extremely rare circumstances a normal person who possesses magic circuits can become a Master.
Regardless of how, if someone makes it to the Moon, clears the preliminaries, and summons and contracts a Servant, they will be able to participate in the main tournament as a Master.

Fate/EXTRA Material: Encyclopedia of Fate/EXTRA