Tybalt has been looking for Romeo ever since he discovered Romeo had attended Capulet's party. Indeed, his enmity against the Montagues is so great that, upon learning of Romeo's presence in the party, he had wanted to kill him even then, even in full view of Capulet's guests.
In this, Tybalt, more than anyone else in the two families, incarnates the feud between the Montagues and the Capulets and the hatred that drives it. You can see this even in his introduction in act 1, scene 1, with his insistence on dueling Benvolio. He hates the Montagues intensely, to such a degree that his entire characterization in the play seems predicated upon that fierce all-consuming hatred he holds for their family, and upon discovering Romeo's intrusion into the Capulet party, he seems to have fixated that intensity of loathing upon Romeo himself. He takes Romeo's presence as an insult, and that insult is made all the worse by his inability to kill Romeo right then and there. Thus, he becomes fixated on dueling Romeo and avenging this slight.
Things come to a head in act 3, scene 1, where Tybalt encounters Mercutio, Benvolio, and Romeo, demanding his duel with Romeo. When Romeo attempts to avoid bloodshed, Mercutio duels Tybalt instead and is killed by Tybalt. With Romeo enraged by the death of his friend, Tybalt finally has his duel and is killed as a result.
Romeo is a Montague, and Tybalt is a Capulet. Early in the play, the audience learns that these two families have a long-standing and bloody feud; much tension and conflict is generated from this "ancient grudge."
Romeo is fairly drug to a party which Lord Capulet is hosting; his friends are attempting to get him to think of something else besides Rosaline, who has recently rejected him. Once he arrives at the party, Tybalt spots him and is filled with rage that Romeo would have the audacity to show his face at a Capulet gathering.
Lord Capulet himself isn't nearly as bothered by Romeo's appearance. In fact, he tells Tybalt that Romeo seems like a "gentleman" and that "Verona brags of him / To be a virtuous and well-governed youth." He instructs Tybalt to be a gracious member of the Capulet family and to entertain their guests, avoiding a scene over a young man who isn't causing any trouble. Tybalt isn't so easily swayed. Viewing Romeo as a "villain," he swears that he will "not endure him."
Tybalt complies with Lord Capulet's request at the party itself, but he is determined to have his revenge for this unwanted intrusion—and as soon as possible. Thus, he finds Romeo in act 3, scene 1, and Tybalt's first words to Romeo again refer to him as a "villain." Tybalt believes that Romeo has inflicted "injuries" upon him and the Capulet family, and he will not be deterred in his quest for revenge.
Short Answer: Tybalt is looking for Romeo to get revenge on him for attending Lord Capulet's party without being invited.
In act 1, scene 5, Romeo, Benvolio, and Mercutio attend Lord Capulet's ball disguised in masks like the other guests in attendance. When Romeo initially sees Juliet for the first time, he is overcome with emotion and gives a moving speech highlighting Juliet's physical attributes. Tybalt recognizes Romeo's voice and threatens to kill him:
This, by his voice, should be a Montague.— (to his PAGE) Fetch me my rapier, boy.—What, dares the slave Come hither, covered with an antic face, To fleer and scorn at our solemnity? Now, by the stock and honor of my kin, To strike him dead I hold it not a sin (Shakespeare, 1.5.52-58).
Tybalt is a Capulet, which means that he is sworn enemies with any member of the Montague family. Because Romeo is a Montague attending a Capulet party without invitation, Tybalt immediately takes offense and believes that Romeo is there to "fleer and scorn" their celebration. Fortunately, Lord Capulet intervenes and prevents Tybalt from attacking Romeo at the party. Later on, Tybalt searches the streets of Verona for Romeo and attempts to fight him for attending Lord Capulet's party.
In Act III, Scene 1 of Romeo and Juliet, Tybalt is looking for Romeo to fight him for attending the Capulet ball uninvited. Because the Capulets and Montagues are in a family feud, Romeo “crashing” the party is disrespectful and forbidden. When Tybalt sees Romeo on the street, he calls him out as being a “villain.” Little does Tybalt know that Romeo has already married Juliet, and they are now relatives. Because of this, Romeo doesn’t want to fight Tybalt, but he can’t tell him why. Mercutio, a hotheaded friend of Romeo’s, says he will fight Tybalt. They start to sword fight, and when Romeo attempts to break up the fight, Tybalt stabs Mercutio. Tybalt and his men run off, but come back still enraged at Romeo. Because Romeo has just lost his best friend, he fights and kills Tybalt. Now, Romeo has killed a relative by marriage and must flee Verona.
This scene is just another episode that seals Romeo and Juliet’s fate of bad luck keeping them from ever getting together.
Tybalt is looking for Romeo because he had challenged Romeo to a duel in revenge for Romeo’s appearing at the Capulet’s ball at the beginning of the play. This is the ball where Romeo meets Juliet.
In Act II, Scene 4, Tybalt is looking for Romeo as he wants to challenge him to a duel. Tybalt has vowed to seek revenge on Romeo for coming to a Capulet party without being invited.