Other than the grudge between the Capulets and Montagues, there's no real reason for Tybalt's aggressive behaviour. It seems possible that he's eager to fight because he wants to defend his reputation as the toughest of the Capulets. It's also likely that Tybalt just likes to fight.
Tybalt was offended by Romeo's intrusion at the Capulet Party, and he was ready to fight him then and there. When Tybalt challenges Romeo in the streets of Verona, Mercutio instead takes the challenge to protect Romeos confusingly well natured advances. Tybalt fataly wounds Mercutio and then he gets hunted down and killed by an inraged Romeo.