challenge him to a duel for attending the Capulet party without an invite.
Tybalt wants to fight Romeo... :)
Tybalt wants to confront Romeo because he feels insulted by Romeo's presence at the Capulet party. He sees Romeo as a Montague intruding on Capulet territory. Tybalt's pride and loyalty to his family drive him to seek revenge against Romeo.
Tybalt reacts angrily and wants to confront Romeo for being at the party. He feels insulted that a Montague would show up at a Capulet gathering and sees it as a provocation.
This would be Tybalt. Of course Capulet is really only his uncle by marriage. Tybalt is Lady Capulet's brother's son. Tybalt does not actually plan to ask for Capulet's permission to throw Romeo out; he calls for his rapier without consulting Capulet first. It is only when Capulet asks him what the heck he thinks he's doing that he explains himself to Capulet.
Capulet's Ball is open to everyone except the Montagues (the family which has a traditional enmity with the Capulets). When Tybalt (a Capulet) realises that Romeo (a Montague) is at the Ball, he wants to start a fight. But Old Capulet (Juliet's father) tells Tybalt that it is a party, and that Romeo seems a nice enough fellow. Tybalt is very angry that he has been prevented from starting a fight. But it isn't so difficult to work out who Romeo is. At the end of the party, when the guests are starting to leave, Juliet asks her nurse to find out who the young man was that Juliet has been talking to. It takes the nurse a few moments to ascertain that it is Romeo - but only a few moments.
Capulet responds by instructing Tybalt to leave Romeo be and not cause trouble during the party. He wants to keep the peace and avoid any conflict.
tybolt
In this scene Tybalt seeks Romeo because he is mad at Romeo for being at the party at the Capulet house, and wants revenge. In the play, this is the only reason but in the movie, Tybalt is mad at Romeo because he knows Juliet is falling for him.
He gets all fired up. He wants to go confront Romeo and fight him for coming to a Capulet party, but Lord Capulet tells him to shut up and be nice. He doesn't want the Prince coming and ending people for starting a fight.
Tybalt, Juliet's cousin, wants to attack Romeo at the Capulet ball because he is angered by Romeo's presence as a Montague. Tybalt is a hot-headed and vengeful character who holds a grudge against the Montagues.
It is actually Tybalt who informs Lord Capulet of the unwanted guest. Tybalt wants to kill him but Lord Capulet orders him not to make a scene at the party because, despite there enmity, Romeo is well liked in the community and it would tarnish his reputation.
Tybalt vows his uncle, Lord Capulet, by expressing his anger and dishonor at Romeo's presence at the Capulet ball and insisting that he wants to challenge him to a duel. He feels that Romeo's intrusion is a personal insult to his family's honor and is determined to seek revenge.