We first meet Juliet's cousin Tybalt in Act 1 Scene 1 when he joins in the street fight with Benvolio and the Capulet and Montague servants. This is a brief appearance, followed by another cameo of sorts in Act ! Scene 5 at the Capulet feast.
In both early scenes, Tybalt is revealed to be loud and quick-tempered, handy with his wit and his sword. Drawn as such, Tybalt is a very useful character for Shakespeare, who uses other characters to bring the lovers together. At the Capulet feast, Tybalt recognizes Romeo's voice and swears vengeance against him for crashing the party.
When Tybalt encounters Mercutio in Act 3 Scene 1, he (Tybalt) has been looking for Romeo to answer a challenge. Romeo appears but will not fight Tybalt, who is now his cousin through marriage to Juliet. Mercutio is enraged by Romeo's cowardice and fights Tybalt himself, gets killed, and prompts Romeo to kill Tybalt. This turns out to be the climax of the play; once Romeo kills Tybalt, there is no chance for the lovers to be together.