Weaver was born in Shrewsbury. His father, a dance teacher, suggested he go to London and become a ballet master. Weaver soon became a specialist in comic roles and created the first pantomime ballet, the burlesque Tavern Bilkers (1702). His more serious work, The Loves of Mars and Venus dealt with themes from classical literature and required a significant amount of gestures due to the story not being expressed in any spoken form. Because Weaver attempted to use plot and emotion in replacement of more sophisticated technical and speech methods, he is considered a major influence on subsequent choreographers, including Jean-Georges Noverre and Gasparo Angiolini.
His books include a translation of Raoul Auger Feuillet's Orchesography, and two original works titled A Small Treatise of Time and Cadence in Dancing (1706) and A collection of ball-dances performed at court (1706).