An act was a portion of a play that was composed of scenes.
In 2153, Drennik quoted act one, scene five of Hamlet - "There are more things in heaven and earth than are dreamt of in your philosophy."- to Jonathan Archer, who was impressed by the Vissian's ability to quote something he had only read yesterday. (ENT: "Cogenitor")
In 2266, Lenore Karidian and her father were reminded that the second act of Hamlet was set to begin in three minutes. She assured the person who reminded them that they'd be ready. (TOS: "The Conscience of the King")
In 2286, expressing his doubts about Spock's memory as the Vulcan had calculated the variables for their time travel from memory, Leonard McCoy quoted, "Angels and ministers of grace, defend us,", which Spock identified as coming from, Hamlet, Act one scene four. This seemed to satisfy the question in James T. Kirk's mind. (Star Trek IV: The Voyage Home)
In 2369, during the rehearsal of her play Something for Breakfast, Beverly Crusher asked William T. Riker to begin at the start of Act two. He began reading Data's poem Ode to Spot instead, due to a malfunction of the USS Enterprise-D's computer. (TNG: "A Fistful of Datas")
In 2371, while discussing the William Shakespeare play Julius Caesar with Julian Bashir, Garak noted that he knew Brutus was going to murder Caesar after reading the first act, something which made the play a farce rather than a tragedy. (DS9: "Improbable Cause")