Tybalt is part of the younger generation of Capulets to continue the feud which has gone on for years and years. Incited by the opening street confrontation between the two families, Tybalt is further angered when he learns that Romeo and friends have come uninvited for his close cousin, Juliet.
Now, in Act III Tybalt wants to know why Romeo was at the party and ascertain his motives, so he calls out in a taunting manner to Mercutio
Mercutio, thou consort'st with Romeo--
When Romeo does arrive, Tybalt insults him,
Romeo, the hate I bear thee can afford
No better term than this--thou art a villain. (III,i,50)
Tyblat further tells Romeo that he has caused him "injuries," and challenges him to a duel. For, the effrontery that Romeo and his friends have displayed by coming to their enemy's house uninvited is considered a grave insult to the Capulet name. In the Italy of the fourteenth century, family name and its honor were of paramount importance.