Why did Tybalt kill Mercutio in the play Romeo and Juliet?

The answer to this question can be found in Act III, Scene 1 of Romeo and Juliet. Tybalt, a Capulet, was not looking for a fight with Mercutio, who is Romeo's friend and not a Montague. Tybalt is searching for Romeo when he encounters Mercutio and Benvolio. When Romeo arrives, Tybalt says "peace be with you" to Mercutio, who has repeatedly tried to goad him into a duel by insulting and challenging him. When Romeo refuses to fight Tybalt, Mercutio steps in and offers a direct challenge that Tybalt cannot refuse with honor. He seems to be holding his own with Tybalt (who he has previously characterized as a master duelist) until Romeo steps in and attempts to restrain him. When he does this, Tybalt runs Mercutio through, giving him a mortal wound. This is a pivotal event because Romeo responds by fighting with Tybalt when he returns. He kills Tybalt, and is banished as a result. This begins the chain of events that lead to his and Juliet's tragic end.