The conflict that is presented in Act III scene 1 of this tragedy is actually rather complex and works on a number of different levels. Firstly, there is the obvious external conflict between the Montagues and the Capulets as Tybalt tries to pick a fight with Romeo. Whilst this is indicated by the obvious insults between the two groups, what is far more interesting is the dramatic irony in this scene: the audience knows what Mercutio and Tybalt do not know, which is that Romeo is actually married to Tybalt's cousin, Juliet, and therefore this explains his reluctance to fight and the following quote:

Tybalt, the reason that I have to love thee
Doth much excuse the appertaining rage
To such a greeting. Villain am I none.
Therefore, farewell, I see thou knowest me not.

The irony in these words is of course true: Tybalt does not "know" or recognise Romeo as a kinsman, and Mercutio as well finds these words and Romeo's inaction in the face of being insulted incomprehensible. The conflict between Mercutio and Tybalt, which leads to Mercutio's death, triggers a massive internal conflict within Romeo, as he then has to choose between satisfying his own rage and avenging the death of his friend, which he is honour bound to do, or respecting the secret ties that bind Tybalt to him as a member of his family. He is really stuck between a rock and a hard place, as there are massive consequences whatever he does, and none of them are favourable. His eventual decision to satisfy his inner rage and avenge Mercutio is incredibly tragic, as it leads to yet further conflict: Romeo is now in conflict with the Prince and the State of Verona, and must leave. Conflict in this scene therefore works on many different levels.