Tybalt is Juliet’s older cousin, head of the younger Capulet men, and the most honour-obsessed person in Verona. Mercutio nicknames him “the Prince of Cats,” after a stock villain in fencing manuals — partly a compliment to his swordwork, mostly a sneer at how seriously he takes himself. Tybalt has no interest in love, no interest in peace, and apparently no interest in anything except being respected.
What Tybalt wants is for Capulet honour to be defended. He shows up in Act 1 Scene 1 itching to fight Benvolio, who’s trying to break up a brawl. He wants Romeo dead from the moment he spots him at the Capulet party — not because Romeo has done anything to him personally, but because Romeo is a Montague at a Capulet party. The grudge is the point.
Tybalt doesn’t change either, and like Mercutio he doesn’t survive his stubbornness. Even when Capulet himself tells him to leave Romeo alone at the ball, Tybalt promises revenge. He challenges Romeo by letter, finds him in the street the day after the wedding (which he doesn’t know happened), and dies for it. The play uses him to make a hard point: someone has to decide to stop, or this is how it ends.