Tybalt becomes upset when he recognizes Romeo at the Capulet party, as he sees it as a direct challenge to his family's honor. Capulet responds by scolding Tybalt for causing a scene and tells him to calm down and let Romeo enjoy the party, not wanting any disruption to the event.
Nobody sent Tybalt. Tybalt sent Tybalt. Why? Because he's decided he doesn't like Romeo and Capulet stopped him from fighting at the Capulet party in Act 1 Scene 5. Tybalt has decided to take matters into his own hands by issuing the challenge.
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.
Romeo and Juliet meet and fall head over heels in love with each other. Tybalt recognizes Romeo and starts to confront him; instead, Lord Capulet restrains Tybalt.
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.
In Act I Scene 5 Tybalt tells Capulet that Romeo, a Montague, has crashed the party. Capulet tells Tybalt not to bother about it: Romeo has a good reputation, and is behaving himself, so why disturb the party at which everyone is having a great time?
Well, it's Tybalt basically, and he is not so much after Montagues as after one Montague, namely Romeo. "Here comes my man" he says. The interesting thing is that Tybalt is not a Capulet. His auntie is married to a Capulet is all. He has no stake in the feud--he just likes to fight. Mercutio isn't a Montague either, since he is related to the Prince. And if you notice, during the trial scene at the end of act 3 scene 1, it is Mrs. Capulet, who is only a Capulet by marriage, who is trying to stir up the feud again, since Tybalt is her relative. Capulet hangs back, because Tybalt is not his relative and he doesn't like him much (think about how he talked to him in Act 1 Scene 5).
The Capulet's ball was to take place at night
Tybalt recognizes Romeo's voice at the Capulet's feast and is outraged that he has crashed the party. He informs Lord Capulet about Romeo's presence and insists that he should be removed from the party immediately. Lord Capulet, however, allows Romeo to stay because he does not want to cause a scene at the feast.
Tybalt sends for his rapier, a kind of sword, in Act I Scene 5 so he can have a fight with Romeo. Capulet forbids it, fortunately.
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.
He believes it will help her feel better after the murder of her cousin, Tybalt.
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.