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.