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.