When Romeo refuses to fight in Act III, scene i, Mercutio thinks that Romeo is being a coward. Tybalt had earlier sent a "challenge" to Romeo's house, and Mercutio expects Romeo and answer it. When Tybalt goes further in this scene and calls Romeo a "villain" (III.i.64), Mercutio expects Romeo to stand up for his own honor and defend himself.
Instead, Romeo, who has just married Juliet, does everything he can to avoid a fight. He says in response to Tybalt's name-calling:
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 (65-68).
The fiery Tybalt, however, refuses to back down, and he urges Romeo to "turn and draw" (70). Romeo again tries to calm Tybalt and assuage his anger:
I do protest I never injured thee,
But love thee better than thou canst devise
Till thou shalt know the reason of my love;
And so, good Capulet, which name I tender
As dearly as mine own, be satisfied (71-75).
Neither Tybalt nor Mercutio knows about Romeo's love for Juliet, much less his marriage to her, so neither can understand why he would possibly hold the name Capulet "as dearly as (his) own." Tybalt is left to stand confused, but Mercutio is aghast at Romeo's behavior, calling it "calm, dishonorable, vile submission" (76).
At this point, Mercutio steps in to defend Romeo's honor, which Romeo seems unable or unwilling to do himself, and the fight is on between Mercutio and Tybalt.