Tybalt plays a significant role as a rather minor character. His quick-temperedness at the party foreshadows the duels in Act III, scene 1. As Lord Capulet rebukes him, one can sense the rage and desire to regain his pride, seething just below the surface. Of course, he challenges Romeo to a duel, but Mercutio steps in instead. Tybalt slays him, but seemingly by accident. Romeo pursues him, and kills Tybalt in a heated battle. This leads to Romeo's banishment.
All of these are instances in which Tybalt affects the tragic nature of the play. Yet let's not forget Juliet, her parents, and the Nurse. Tybalt's death affects each of them, driving the tragic plot further. For Juliet, it forces her to reconsider her love for Romeo. Her parents drive her into marriage with Paris, thinking it is her sadness over her cousin's death that keeps her so melancholy. And the Nurse turns her back on Romeo, seeing him only as a murderer and a disdainful man. Thus, Tybalt is quite influential throughout the play.
He is a nephew to Lord Capulet and a cousin to Juliet. He does not speak many lines, but he influences the entire course of the play to a degree that exceeds his seemingly minor role in it. Throughout the play, he demonstrates his angry, resentful, and stubborn nature. Tybalt urges on the fight of the Capulet and Montague households. In addition to his being belligerent and stubborn, Tybalt also has no qualms about fighting unfairly.
How does the secondary character you have chosen influence Romeo and/or Juliet?
Tybalt is Juliet's cousin. He's hot-headed and hates Montagues. If not for him, Mercutio wouldn't have died (seeing how he killed him) which means that Romeo wouldn't have been banished (seeing how he killed Tybalt). Now if he wasn't banished, there would be no reason for the Friar to send a message to Mantua. Which means that he would have known that Juliet's "death" (if that was even still played out) wasn't real. Which means that he wouldn't have killed himself and she wouldn't have killed herself. ^-^