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.
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.