In act 2, scene 4, Mercutio enters and demands to know whether Romeo is likely to return home that day. At this juncture, Mercutio still thinks that Romeo is besotted with his previous love, Rosaline, and fears that his obsessive interest in her might have led him into dangerous situations. According to Benvolio, Romeo hasn't been home in a couple of days; when Benvolio tells Mercutio that Tybalt has sent a letter to the house of Romeo's father, Mercutio immediately fears that it is "a challenge" and notes that Romeo, especially in his current lovesick state, is really no match for Tybalt. Mercutio describes Tybalt as "more than a prince of cats," an excellent fighter, and somebody who Romeo could not easily stand against even if he were not already "dead"—that is, in a lessened state because of his lovesickness. After hearing of the letter from Tybalt, Mercutio is even more eager to find Romeo, concerned about what he may be up to. When Romeo appears, he is greatly relieved and accuses Romeo of having given them "the slip" last night.

As Act II, Scene 4 of Shakespeare's Romeo and Juliet opens, Benvolio and Mercutio are speaking to each other. Mercutio is concerned about Romeo because of his friend's unrequited love for Rosaline. Benvolio reveals that Romeo didn't go home the night before and he also tells Mercutio that Romeo has received a letter from Tybalt. Tybalt is eager to challenge Romeo because he feels that Romeo has crashed Capulet's party "To fleer and scorn at our solemnity." He wants to immediately confront Romeo as soon as he hears his voice. Lord Capulet, however, refuses to allow Tybalt to disrupt the proceedings and even gives Romeo a compliment by saying that "Verona brags of him/To be a virtuous and well-governed youth." Tybalt is incensed and, while he stays quiet at the party, vows to go after Romeo later:

Patience perforce with willful choler meeting
Makes my flesh tremble in their different greeting.
I will withdraw, but this intrusion shall,
Now seeming sweet, convert to bitt’rest gall.

BEN:
Tybalt, the kinsman to old Capulet,
Hath sent a letter to his father's house.

MER:
A challenge, on my life.

BEN:
Romeo will answer it. 

Romeo has been spotted by Tybalt at the Capulet party (Act 1, Scene 5). Tybalt, remember, is furious that Romeo has dared to show his face at a Capulet party (Romeo is a MOntague, a family who Tybalt hates). Lord Capulet tells Tybalt to calm down and stop fussing - and then, when Tybalt continues, bans him from saying anything else.

Tybalt has sent a formal challenge to Romeo's father - wanting to fight him for going to the Capulet party. Of course, Romeo does not answer it - though Tybalt does appear himself in Act 3, Scene 1, to fight Romeo. And he actually ends up fighting Mercutio. And the rest is ...well, tragedy.

Hope it helps!