There are six major genres of literature: fiction, nonfiction, short story, poetry, drama, and folktale/mythology. Within these genres are many subgenres, such as romance, fairytale, comedy, tragedy, and the like.
Shakespeare's play "Othello," simply because it is a play, falls into the genre of drama. More specifically, it belongs in the subgenre of tragedy. The Guide to Literary Terms defines tragedy as
a serious play in which the chief figures, by some peculiarity of character, pass through a series of misfortunes leading to the final catastrophe. In contemporary theater, tragedy often has the evils of society as the cause of this downfall.... In literature, tragedy refers to any composition with a somber theme carried to a disastrous conclusion....
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"Othello" is a tragedy, and along with "Hamlet", "Macbeth" and "King Lear", considered among the four "great" tragedies of Shakespeare.
Unlike some of the tragedies though, there is little sense of the supernatural about Othello: Iago deliberately plans and executes the events which lead from Othello's suspicion to Desdemona's murder.