Tybalt is Juliet's cousin, which makes him a Capulet. After he kills Romeo's BFF, Mercutio, in a street brawl, Romeo mortally stabs him, which causes Romeo to be banished from Verona.
Tybalt is a captivating, testosterone-driven character and almost always completely over-the-top. He's not particularly deep, but he's a lot of fun for the actor who gets to deliver his snappy one-liners and show off some impressive sword fighting skills. Mercutio, who hates Tybalt, gives him the "catty" nickname the "Prince of Cats" and it totally suits Tybalt. While Romeo can sometimes remind you of a bouncy and overeager puppy, Tybalt tends to stalk around proudly looking for fights. When his uncle Capulet prevents him from beating up Romeo for crashing the Capulet's masked ball, he's not too pleased and promises to bash in Romeo's skull at a later date: "I will withdraw but this intrusion shall / Now seeming sweet, convert to bitt'rest gall" (1.5.6). Clearly, Tybalt, likes to speak in rhymed couplets ("shall" and "gall" rhyme here), which makes him sound kind of ridiculous. Plus, he doesn't speak a single line that can't be delivered in a snarl.
Aside from the vendetta between the Capulets and Montagues, there's no real explanation for Tybalt's aggressive behavior. 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, which brings us to our next point. If there's a personification of hate in the play, it's Tybalt. Think, for example, of the fact that while super macho Tybalt is storming around the Capulet ball threatening to beat Romeo to a pulp (just for being a Montague), Romeo and Juliet are a few feet away being all sappy sweet and professing their love for each other (1.5). In fact, Romeo and Juliet's first encounter occurs on the heels of Tybalt's thwarted rampage.