Tybalt is nephew to Lady Capulet, cousin to Juliet, and the house's duelist, and he calls Capulet "uncle." In Act III, scene 1, he kills Mercutio and is killed by Romeo, and it is this death that leads to Romeo's banishment.
In Act I, scene 1, he arrives at the street brawl, challenges Benvolio, who only wants to keep the peace, and says he hates the word peace as he hates hell. At the ball (scene 5), he recognizes a Montague by his voice and asks for his sword to kill him on the spot. Capulet restrains him and calls him an impudent boy, and Tybalt leaves swearing that the intrusion, sweet in appearance, will turn to gall. In Act II, scene 4, he sends a challenge to Romeo's house.
In Act III, scene 1, he looks for Romeo in the square and calls him a villain; Romeo refuses the duel and says that he loves him more than Tybalt can imagine. Mercutio takes up the challenge for him, and Tybalt wounds him under Romeo's arm and flees. He returns soon after, with Mercutio dead, and Romeo kills him.
Tybalt's death shapes the rest of the play. Juliet mourns her cousin, and the Nurse calls him "the best friend I had" (Act III, scene 2). Capulet arranges Juliet's marriage to Paris in a small feast, out of respect for the dead (scene 4). Juliet sees his ghost before drinking the potion (Act IV, scene 3), and Romeo, in the tomb, asks his forgiveness (Act V, scene 3).