What is the meaning of 'catharsis' in a tragedy?

First of all, I'm aware of the Greek origin of the word. I have looked this word up in a dozen different dictionaries but I still don't quite understand what a catharsis is. A definition:

Originating from ancient Greece, the word "catharsis" refers to the emotional outpouring of a character. Often, when a character in a tragedy realizes his/her flaws or downfall, a cathartic speech is delivered.

In other definitions it also says that the audience also experiences this, but I don't understand at all what is meant by this. The above definition reminds me of an epiphany, but I don't see at all why the audience would experience this too, since the flaws of a character are usually known by the audience from the start. Can someone clear up my confusion regarding this subject?

The short answer is No, someone can’t clear up your confusion. Scholars of Greek philosophy and of dramatic criticism have been arguing about this since at least the middle of the 18th century and have come to no firm conclusion.

The slightly longer answer is that the term derives from Aristotle’s Poetics, in his definition of tragedy as

“an imitation of an action that is serious, complete, and of a certain magnitude; in language embellished with each kind of artistic ornament, the several kinds being found in separate parts of the play; in the form of action, not of narrative; through pity and fear effecting the proper catharsis of these emotions.

And that’s all he says about catharsis.

Aristotle uses the term in other works in the medical sense of purgation — so the metaphor here implies that somehow somebody is being purged of “unhealthy” pity and fear. And it’s generally conceded that he’s confronting Plato’s very negative opinion of mimetic poetry as an exciter of ‘base’ emotions and impulses —

lust and anger and all the other affections [...] desire and pain and pleasure, which are held to be inseparable from every action —in all of them poetry feeds and waters the passions instead of drying them up; she lets them rule, although they ought to be controlled, if mankind are ever to increase in happiness and virtue.

So Aristotle is presumably claiming that tragedy arouses pity and fear in order to ‘purge’ them and restore emotional balance to the hearers. But the mechanics of this operation are entirely conjectural.

There's a respectable article on Wikipedia which can usefully launch you into a deeper dive.

EDIT:
By the way, the notion of catharsis operating on characters is not widely held; and the notion of a cathartic speech in drama — basically employing the psychological sense which Kristina Lopez’ answer advances — is absent from Aristotle and from LitCrit discourse.

Definitions are great, but let's use an example of catharsis:

Your dog dies. Of course you miss him but you can't seem to actually grieve him or even shed a tear. Instead, you get mad every time you see his favorite red ball in the yard and you tell your friends to bug off when they mention the dog.

A couple weeks later you pass a pet shop and see puppies playing in the window. The grief you've surpressed overwhelms you and you find yourself sobbing uncontrollably in front of that store window as you feel the tremendous loss of your canine companion. Feeling foolish, but surprisingly relieved, you can now talk about your dog with friends and can start to heal your broken heart because of the catharsis you experienced that day at the pet store.

Does that help explain what it is?