Mimesis - Mimesis is the act of creating in someone's mind, through artistic representation, an idea or ideas that the person will associate with past experience. Roughly translatable as "imitation," mimesis in poetry is the act of telling stories that are set in the real world. The events in the story need not have taken place, but the telling of the story will help the listener or viewer to imagine the events taking place in the real world.
Hamartia - This word translates almost directly as "error," though it is often rendered more elaborately as "tragic flaw." Tragedy, according to Aristotle, involves the downfall of a hero, and this downfall is effected by some error on the part of the hero. This error need not be an overarching moral failing: it could be a simple matter of not knowing something or forgetting something.
Anagnorisis - This word translates as "recognition" or "discovery." In tragedy, it describes the moment where the hero, or some other character, passes from ignorance to knowledge. This could be a recognition of a long lost friend or family member, or it could be a sudden recognition of some fact about oneself, as is the case with Oedipus. Anagnorisis often occurs at the climax of a tragedy in tandem with peripeteia.
Mythos - When dealing with tragedy, this word is usually translated as "plot," but unlike "plot," mythos can be applied to all works of art. Not so much a matter of what happens and in what order, mythos deals with how the elements of a tragedy (or a painting, sculpture, etc.) come together to form a coherent and unified whole. The overall message or impression that we come away with is what is conveyed to us by the mythos of a piece.
Katharsis - This word was normally used in ancient Greece by doctors to mean "purgation" or by priests to mean "purification." In the context of tragedy, Aristotle uses it to talk about a purgation or purification of emotions. Presumably, this means that katharsis is a release of built up emotional energy, much like a good cry. After katharsis, we reach a more stable and neutral emotional state.
Peripeteia - A reversal, either from good to bad or bad to good. Peripeteia often occurs at the climax of a story, often prompted by anagnorisis. Indeed, we might say that the peripeteiais the climax of a story: it is the turning point in the action, where things begin to move toward a conclusion.
Lusis - Literally "untying," the lusis is all the action in a tragedy from the climax onward. All the plot threads that have been woven together in the desis are slowly unraveled until we reach the conclusion of the play.
Desis - Literally "tying," the desis is all the action in a tragedy leading up to the climax. Plot threads are craftily woven together to form a more and more complex mess. At the peripeteia, or turning point, these plot threads begin to unravel in what is called the lusis, or denouement.
According to Wikipedia.com, hamartia is a term that was developed by Aristotle and:
...can simply be seen as a character’s flaw or error...hamartia is the tragic flaw of the protagonist in a given tragedy.
The word hamartia is rooted in the notion of missing the mark (hamartanein) and covers a broad spectrum that includes accident and mistake, as well as wrongdoing, error, or sin.
Shakespearean tragedies come to mind, such as Hamlet and Macbeth. Shakespeare's tragic heroes all had a tragic [character] flaw. For Hamlet it was indecision. For Macbeth it was vaulting ("blind") ambition.
....misfortune is not brought about by villainy but by some “error of judgment” (hamartia). This imperfection later came to be interpreted as a moral flaw, such as Othello’s jealousy or Hamlet’s irresolution...
The basis for understanding hamartia, then, is that a mistake is made, but it is due more to an error in judgment, or, more typically perhaps, a flaw in one's character, rather than springing from malicious intent. It was something created by Aristotle. Catharsis is also something connected to the Greek, and it also was used by Aristotle.
Catharis means "cleansing" or "purging," but it is not necessarily referred to literally. One may experience a catharsis by experiencing a bout of prolonged weeping. Screaming or yelling, or even unburdening one's guilt can be cathartic. The idea is that a weight is lifted off of one's heart or soul.
The emphasis of "catharsis" as an emotional response was introduced by Aristotle as well.
The Greek philosopher Aristotle was the first to use the term catharsis with reference to the emotions—in his work Poetics. In that context, it refers to a sensation or literary effect that, ideally, would either be experienced by the characters in a play, or be wrought upon the audience at the conclusion of a tragedy; namely, the release of pent-up emotion or energy.
Therefore, in a tragedy, hamartia refers to a hero's tragic flaw, which drives him to do things that not only affect those around him, but ultimately his own fate as well.
Catharsis is something that can also be seen in a tragedy. For instance, when Claudius and Macbeth, two of Shakespeare's great villains, die (in Hamlet and Macbeth, respectively), it may be cathartic for the audience, seeing such heinous criminals punished for their horrific deeds. When Hamlet kills Claudius, we may assume it is cathartic for him as well, although his tragic flaw (harmatia) has already sealed his own fate.