A pun, or a play on words, is where one word is deliberately taken to mean something else and a comic effect is produced as a result. We can see a pun at work in the conversation, or barrage of insults, that come into play after Baptista has made his decree that Bianca will not marry until Katharina is married. In this case, the pun centres around the use of the word "mates." Trace the conversation between Katharina and Hortensio:
KATHARINA: I pray you, sir, is it your will
To make astale of me amongst these mates?
HORTENSIO: "Mates," maid? How mean you that? No mates for you,
Unless you were of gentler, milder mould.
Katharina uses the word "mates" to refer to rude fellows, deliberately insulting Gremio and Hortensio, but Hortensio deliberately twists what she is trying to say, using "mates" in the sense of refering to "husband."