Analysis: What's Up With the Title?
Everybody knows The Taming of the Shrew is not about the domestication of an actual "shrew" (the mammal Sorex araneus). Rather, the play's title refers to the way Petruchio breaks the will of the "shrewish" Kate – a stereotypically bossy, mouthy, and aggressive woman who is seen as trying to "wear the pants" in her relationships with men.
The term "shrew" was pretty common in Elizabethan England and it was applied to just about any woman who stepped outside of her prescribed social role. The term was also somewhat synonymous with the word "scold," a legal term used to describe disruptive women (sometimes men too). It's interesting to note that the term "shrew" is also applied to Petruchio when describing the way he treats his wife. The idea is that Petruchio acts like a "shrew" in order to give Kate a dose of her own medicine.
"Taming" is an apt description of Petruchio's behavior – it's a term he and other men use throughout the play to describe the process of breaking the will of an unruly wife. Kate is treated very much like an animal – at one point, Petruchio refers to her as a falcon and compares his torture and abuse of Kate to the process of training a wild bird of prey (both must be starved and deprived of sleep until they willingly obey their masters' commands).
A final note. The Taming of the Shrew shouldn't be confused with The Taming of a Shrew, a play written around the same time. Both are quite similar in content but A Shrew is inferior to Shakespeare's text – it's likely a pirated version of our boy's work.