Katherina is obviously the "shrew" referenced in William Shakespeare's The Taming of the Shrew, and she lives up to that reputation, at least in most ways. A shrew is defined as a "an unpleasant, bad-tempered woman" or a "an ill-tempered scolding woman." Either one is bad, and Katherina seems to be both.
The first time we meet her she is described this way:
"That wench is stark mad or wonderful froward."
Katharina throws things, hits people, and screams a lot, and most of the people in her life are at least moderately afraid of her--most are flat-out terrified of her temper and her sharp tongue. She is the talk of the town because everyone knows what kind of person she is, and they also know that her father is afraid he will never be able to marry her off. That is evident by the fact that he has to resort to bribery, hoping someone who wants ot marry his younger daughter, Bianca, will be motivated to find someone willing to marry the shrewish Katherina.
That is exactly what happens, of course, and Petruchio is unmoved by her tantrums, determined to "kill her with kindness." Every moment of their relationship seems designed to humiliate, frustrate, and hurt Katherina, and the method is effective until Katherina is eventually "tamed."
Those are the evident and obvious characteristics of Katherina; however, she is a much more complex character than just a raving harpie who is angry at everyone. Based on her sharp-tounged wit, it is obvious that she is a very bright young woman. She is also someone who demands respect, as demonstrated here:
Why, sir, I trust I may have leave to speak;
And speak I will; I am no child, no babe:
Your betters have endured me say my mind,
And if you cannot, best you stop your ears.
My tongue will tell the anger of my heart,
Or else my heart concealing it will break,
And rather than it shall, I will be free
Even to the uttermost, as I please, in words.
It is easy, when reading the play, to assume that Katherina has always just been this way, but it is likely that she has not. Her father has put her on the marriage auction block, so to speak, when he says to two of Bianca's suitors (neither of whom has any interest whatsoever in Katherina, as he well knows):
If either of you both love Katharina,
Because I know you well and love you well,
Leave shall you have to court her at your pleasure.
While it may appear that Bianca is "the good sister," I will remind you that she lies to and disobeys her father at every opportunity when there is something she wants (Hortensio); since she does it so willingly and easily, we must assume that this is a pattern with her. It is easy to see that Bianca has rather manipulated her father into seeing her in a positive light, something Katherina was not willing to do. She is human and she wanted her father's attention as much as Bianca did, but she was not willing to stoop to trickery and deceit to get it like Bianca was. Oddly enough, then, Katherina is the better person, and in the end, of course, Bianca is a much more disobedient wife than Katherina.
Petruchio wins his battle to tame Katherina, but only because she lets him do so. This may be Katherina's story, but she is not the only shrew in the tale. The widow and Bianca both exhibit some pretty shrewish behavior at the banquet--they are harsh, scolding, and ill-tempered. Katherina's descent into "shrew-dom" was a reaction to her circumstances, and her ascent out of "shrew-dom" was a decision she made.