Master is a designation given to a magus who has formed a contract with a Servant. Seven Masters are selected at the start of the Holy Grail War (a member from each of the Three Founding Families will always be selected). Masters are bestowed with Command Spells by the Holy Grail, and earn the right to participate in the Holy Grail War. Each Master has a wish, and pursues the Holy Grail as a means to grant that wish, which drives the fierce battles between Masters (there are exceptions such as Uryuu Ryuunosuke).

Fate/Zero Animation Guide I: Glossary of the holy grail war I

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

The title held by those who previously participated in the Holy Grail Wars.
For the most part, they are Wizards (or, in the days of magic in the physical world, Mages), but sometimes normal people who happen to have magical circuits are registered as Masters
as well.
They form contracts with Servants who will fight on their behalf, and command them in battle. When the new SE.RA.PH was created, the only remaining Master who was present there was the Main Character, but following the ending of EXTELLA, many Masters should begin to appear as new inhabitants.

Fate/EXTELLA material: Encyclopedia of Fate EXTELLA