The conflict between Tybalt and Mercutio is not personal. They fight because the Capulets and Montagues are fighting.
The feud between the Montagues and the Capulets has been going on for a long time. In fact, it has been going on for so long that no one seems to know what started it. When Mercutio and Tybalt get into a duel, they are fighting for honor. As soon as they meet each other on the street, they start insulting each other and it is not long before it turns into a duel, even though fighting has been forbidden by the prince.
Tybalt is angry at Romeo. He feels that he crashed the Capulet ball. Tybalt calls Romeo a villain, but Romeo claims to love him.
Boy, this shall not excuse the injuries
That thou hast done me; therefore turn and draw.
I do protest I never injur'd thee,
But love thee better than thou canst devise (Act 3, Scene 1)
Mercutio stands up for Romeo because he is his friend and Romeo is clearly out of his element here. Mercutio seems to be the better swordsman and is definitely witty. Yet unfortunately Romeo panics and gets in the way. Mercutio is stabbed under Romeo’s arm and dies. Therefore Romeo accidentally causes Mercutio’s death in the middle of the fight between Tybalt and Mercutio.
Shakespeare demonstrates that the conflict between Tybalt and Mercutio grows out of the Montague-Capulet conflict. Tybalt’s problem with Romeo really comes from his being a Montague. Romeo really has no problem with Tybalt. Mercutio doesn’t really like Tybalt much, calling him “ratcatcher” and “King of Cats” to demonstrate this. He stands up for Romeo. Tybalt kills Mercutio because he is fighting him instead of Romeo. The conflict escalates too quickly because of both of their tempers and Romeo’s immaturity in refusing to acknowledge Tybalt and then getting in the way.