In William Shakespeare's Romeo and Juliet, Tybalt is a "kinsman of old Capulet," while Mercutio is one of Romeo's kinsmen and his friends. In the opening scene of Act III, Mercutio and Benvolio (another of Romeo's friends) are out in Verona when they encounter some of the Capulets.
Tybalt wants to talk to the pair, but Mercutio immediately begins with a hostile tone saying, "make it a word and a blow." Romeo comes upon the group and the hostilities escalate. Tybalt soon challenges Romeo to "turn and draw". Romeo, however, does not want to fight with Tybalt. Mercutio, though, does resume a hostile posture and swordplay breaks out between the two despite Romeo's efforts to stop it. In the confusion, Tybalt stabs Mercutio under Romeo's arm ("I was hurt under your arm").
So, it would appear that Tybalt kills Mercutio because Mercutio had acted in a hostile manner toward him and because Tybalt and Mercutio were on opposite sides of the war between the Capulets and the Montagues.