In Romeo and Juliet, why does Shakespeare have Romeo and Tybalt duel? Why does he contrast these two young men in Act II?

The duel between Romeo and Tybalt is required by the story, as outlined by the Prologue, and seemingly demanded by fate.  The Chorus says that "ancient grudge [will] break to new mutiny" and that this will bloody the hands of those who should be civil.  Further, without Tybalt's death, Romeo would never have been banished, and if he were not in exile, he would have known about Juliet's plan, and then the lovers would not...

The duel between Romeo and Tybalt is required by the story, as outlined by the Prologue, and seemingly demanded by fate.  The Chorus says that "ancient grudge [will] break to new mutiny" and that this will bloody the hands of those who should be civil.  Further, without Tybalt's death, Romeo would never have been banished, and if he were not in exile, he would have known about Juliet's plan, and then the lovers would not have a "death-marked love" or be "star-crossed."  Romeo must kill Tybalt in order for fate, and the play's story, to advance. 

It is, ultimately, the feud between the two families that is responsible for the deaths in the play, and Tybalt's death is yet another casualty of the violence and the ridiculous grudge between these families.  When Romeo slays Tybalt, it gives the families another opportunity to address the feud and right their wrongs, but it only makes them more volatile going forward.  Rather than see his death as a reason to appreciate their daughter all the more, the Capulets seem to learn nothing about the brevity or value of life and the terrible cost of violence; they are willing to cast her into the streets if she disobeys them.  Thus, Tybalt's death gives the audience another opportunity to identify who is most at fault for the tragedy: the warring parents.

Shakespeare likely contrasts Tybalt and Romeo to show how warlike Tybalt is compared to how loving Romeo tries to be.  Though Tybalt insults and dishonors him, Romeo swears that he has a secret reason to love Tybalt, and so he refuses to fight.  The violence of the feud has so infected Tybalt's mind that he cannot back down, and he becomes yet another casualty of it.  This, in turn, leads to Romeo's own eventual death.  The violence has such deadly consequences that neither of the families could recognize until it was too late.