Aristotle's Poetics has certainly been very influential in many ways. It not only perfectly defines a tragedy, other authors have used his definitions as a model ever since. Since we are limited to space and won't have room to cover all of his concepts concerning tragedy, below is a discussion of his definition of a tragic hero.
Aristotle saw that a tragic hero must be someone that we respect, admire, and value. If the hero was anyone the reader/viewer disliked, then the story would actually cease to be tragic. The reader/viewer would exult in the fall of a villain. For that reason, Aristotle defines a tragic hero as one who is of high social standing, has a valued reputation, and is generally prosperous, as we see in Aristotle's line, "[a tragic hero] is one of those who stand in great repute and prosperity, like Oedipus and Thyestes: conspicuous men from families of that kind" (as cited in "Aristotle & The Elements of Tragedy"). Not only must the tragic hero be a prosperous person of high social standing, he must also be a generally good and virtuous person. The tragic hero must be someone we admire, otherwise the story can't possibly be tragic. However, the tragic hero must also have a fatal character flaw that leads to his/her demise. In other words, the hero must fall, either through death or some other great loss, due to a choice he/she makes. But the fall will not be completely the fault of the flaw; other circumstances will also have a helping hand, such as fate.
Many authors have modeled their tragic heroes based off of this definition. Shakespeare was especially fond of both Aristotle and the classics and modeled many of his tragic heroes and tragedies after Aristotle's definitions. Romeo in Romeo and Juliet is a perfect example of a tragic hero. Not only is he the son of a lord, giving him high social status, he is a person whom all of Verona admires for being a "virtuous and well-govern'd youth," as his enemy Lord Capulet phrases it to Tybalt (I.v.71). But most importantly, the fatal flaw of being impetuous and acting upon his rash, passionate emotions rather than reason leads to his making poor decisions, resulting ultimately in his death as well as other deaths.