In this lively scene, Romeo runs into Benvolio and Mercutio. Although both friends think Romeo is still pining for Rosaline, he, of course, has fallen in love with Juliet. Therefore, he is in cheerful spirits, no longer moping around. Here we see two wordsmiths—Romeo and Mercutio—cross not words but swords. The puns abound—and we get a deep insight into the friendship that holds these two together.
Mercutio wonders what happened to Romeo after the Capulet party the night before. Mercutio says "you gave us the counterfeit last night." When Romeo asks what counterfeit, Mercutio replies "the slip, sir, the slip." Slip is a pun on Romeo slipping away from them and also slang at that time period for counterfeit money.
Romeo says that in such a "case" as his, he had a right to slip away. Case here means both "situation" and "physical condition," meaning pining both mentally and physically for a woman.
Mercutio catches that lust was part of Romeo's "case" and makes a joke about Romeo having a worn-out "pump," punning on pump as meaning shoe and sexual organ. Romeo responds with the exclamation that this is "single-soled jest," sole a pun on the sole of a shoe and one's soul.
As we can see, the two match each other word for word.
When he meets Juliet's nurse, Mercutio puns on "hare" as meaning whore and then puns on the word "hoar" as meaning both moldly and a whore. He thus implies that the nurse is both old and a whore.
All of these explanations of puns come from footnotes found in the Bevington edition of The Complete Works of Shakespeare.