As a young child, Glover displayed an affinity for rhythms, and at age four he began taking drumming lessons. Deemed too advanced for the class, however, he then enrolled at the Newark Community School of the Arts and soon became the youngest person in the school's history to receive a full scholarship. At age seven he began taking tap lessons and quickly developed a passion for rhythm tap, a form that uses all parts of the foot to create sound. His talent attracted the attention of a choreographer for the Broadway musical The Tap Dance Kid, and Glover served as an understudy before taking the lead role in 1984. He returned to Broadway in 1989, performing in the musical revue Black and Blue, and was nominated for a Tony Award. A role in the motion picture Tap (1989) followed. Glover, who had long made a point of learning as much as he could from old tap masters, soon began teaching tap classes. He also developed his own tap style, which he christened “free-form hard core,” while working with dancers such as Gregory Hines (Hines, Gregory), Henry Le Tang, and Sammy Davis, Jr. (Davis, Sammy, Jr.)

In 1990 Glover created his first choreography, for a festival at New York City's Apollo Theatre. Two years later he became the youngest-ever recipient of a National Endowment for the Arts grant. He portrayed a young Jelly Roll Morton (Morton, Jelly Roll) in the musical Jelly's Last Jam, which debuted in Los Angeles in 1991 before opening on Broadway the following year and touring in 1994. In 1995 Bring in 'Da Noise, Bring in 'Da Funk opened Off-Broadway. Glover choreographed and starred in the musical, which featured a series of vignettes that chronicled African American history. A huge success, the show soon moved to Broadway, and in 1996 it won four Tony Awards, including a best choreographer award for Glover. His numerous other appearances included a regular role (1990–95) on the children's television show Sesame Street. In 2000 Glover appeared in director Spike Lee (Lee, Spike)'s film Bamboozled.