What happens
Benvolio and Mercutio are out in the heat. Tybalt arrives looking for Romeo — he is owed satisfaction for the Capulet party. Romeo enters; Tybalt insults him; Romeo, who has just been secretly married to Tybalt's cousin, refuses to fight. Mercutio, taking Romeo's restraint as cowardice, draws on Tybalt himself. Romeo tries to break it up. Tybalt stabs Mercutio under Romeo's arm, then runs. Mercutio dies, cursing both houses. Romeo, undone, finds Tybalt and kills him. The Prince arrives and, instead of executing Romeo, banishes him to Mantua.
Why it matters
This is the scene Romeo and Juliet stops being a romantic comedy and becomes a tragedy. Up to here you could plausibly believe the play might end with two weddings and a reconciled Verona. After Mercutio dies, you can't. Shakespeare kills his funniest, most life-loving character first, and the play after this is audibly heavier in the language. It's craftwork: the joy goes out of the play with Mercutio.
Romeo's choice in this scene is one of the most argued-over in Shakespeare. He has been married to Juliet for less than a day. He has every reason — the play's best reasons — to keep walking away. He kills Tybalt anyway. The line he says right after — "O, I am fortune's fool" — has him handing his agency back to fate. The play won't let him. Every choice in this scene is his own. He chose restraint, then he chose violence, and the consequences are his.
The Prince's verdict is also a choice. The first scene of the play warned that the next person to brawl would die. The Prince doesn't kill Romeo. He banishes him. Mercy in the abstract, disaster in particular: it's the banishment, not an execution, that requires the Friar's secret-letter plan, that requires Friar John's quarantine, that requires the tomb. The play presses, hard, on how a kindness can make everything worse.