Why does Tybalt hate Romeo?

There are two reasons for Tybalt's hatred of Romeo. First, Shakespeare states in the Prologue that there is an "ancient grudge" between the Montagues and Capulets. In other words, the two families have a long standing feud, which often erupts in the streets of Verona. In fact, the very first scene of the play involves violence in the streets between the warring families. When Tybalt, a Capulet, encounters Romeo's cousin Benvolio, a Montague, he indicates in no uncertain terms his dislike for the Montagues:

What, drawn and talk of peace? I hate the wordAs I hate hell, all Montagues, and thee.Have at thee, coward!

Second, Tybalt is personally insulted that Romeo has come uninvited to Lord Capulet's party in Act I, Scene 5. Even though Romeo wears a mask, Tybalt recognizes him from his voice. He flies into a rage, calling for his sword and intending to fight Romeo on the spot:

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.

Lord Capulet, fearing disruption at what should be a joyous occasion, ultimately persuades Tybalt to stand down. Tybalt, however, vows revenge and sends Romeo a threatening letter, calling for Romeo to meet him for a fight. In Act III, Scene 1, Tybalt gets his chance with unfortunate results for everyone.