Here, a master's is a master's degree: the university award you take after a first degree, usually one or two extra years. People say “I'm doing a master's” or “She has a master's in history.” The little mark (') is part of the spelling. It started as master's degree — the degree of a master — and then people dropped the word degree in speech. It is everyday school-and-work English. On official papers you will see the full name, like MA or MSc. Though actually, master's can also mean “belonging to the master” (“the master's chair”). When people are talking about university, though, they almost always mean the degree.