What is the meaning of the masque and anti-masque in Shakespeare's play The Tempest?

A Renaissance masque was a spectacle performed at court or at the manor of a member of the aristocracy with the purpose of glorifying a particular member of court or a particular aristocrat. In Jacobean times, a masque was an interactive form of elaborate entertainment for the aristocrats. In his plays, Shakespeare often incorporates masques because they were something that everyone—noblemen and commoners alike—were familiar with. It is also worth noting that it was Ben Jonson, in the seventeenth century, who developed the masque and then the anti-masque into a literary form:

While many masques had tended to move in one of two directions, either "wholly literary and dramatic or wholly choreographic and theatrical," Jonson aimed to unify the poetic and spectacular elements into a cohesive whole. Jonson also developed the anti-masque, an addition which enabled him to achieve this cohesive whole. The anti-masque introduces comic or grotesque characters and plot material to act as foils to the main masque or to allow the main masque to provide a resolution (Cunningham 110; Orgel 76). Such a literary convention, the introduction of the evil or grotesque in the anti-masque, served to further enhance the primary purpose of the masque to glorify the court and, in particular, the monarch.

http://www.mith.umd.edu/comus/cegenre.htm

The main components of a literary masque were a poetic induction or prologue, antimasque(s), main masque, revels, and then an epilogue. Now that we know what masques and anti-masques are, we can see how Shakespeare uses them in The Tempest.

The most striking instance of the masque is the engagement of Ferdinand to Miranda in Act IV, Scene I. Prospero conjures a masque at this event and various goddesses perform. However, the masque in this scene has broader implications as it reminds Prospero of the bridge between magic and reality and paves the way for his return to power. The following anti-masque depicts Trinculo, Stephano, and Caliban being chased by dogs because they stole garish costumes. There are also other examples of masques and anti-masques throughout the play.

Of course, the play as a whole can be viewed as a masque—given the elements of a literary masque (described above), perhaps you can see how this might be the case.