For many years, literary critics have agreed with and used Aristotle's definition of a tragic hero found in Poetics to understand what a tragic hero is. Aristotle argued that a tragic hero is a noble person of higher social standing, like a prince or the son/daughter of a lord, who has some character flaw that leads to his/her downfall. The character, in general, is also a virtuous and respected person, so the audience does not feel that the character's fate is deserved, making the death or downfall very painful for the audience.
There are several ways in which Romeo fits this definition of a tragic hero. To start with, we know that he is of high social standing as his father is a lord. He's also recognized as a generally good and innocent character. Even Lord Capulet praises Romeo's character when he is discovered at their ball, using Romeo's praise to argue that Tybalt should not fight him for having crashed the ball. We see Capulet praise Romeo's character in the lines:
Content thee, gentle coz, let him alone.
He bears [himself] like a portly gentleman,
And, to say truth, Verona brags of him
To be a virtuous and well-govern'd youth. (I.v.68-71)
By "portly," Capulet means that Romeo's behavior is "stately" and "dignified," just like a noble lord's son ought to be (Random House Dictionary). Therefore, the phrase "portly gentleman" coupled with the statement about his virtue shows that Romeo is indeed the type of innocent, virtuous character that an audience would sympathize with as a tragic hero.
In addition, Romeo also has character flaws that lead to his death. It is not just fate alone that is responsible for Romeo's death; instead, his choices and actions play just as great a role. One of Romeo's character flaws is that he is very irrational and emotionally driven. We see his irrationality and emotionalism when we meet him in the very first scene. We learn that his parents are concerned about him because he has been staying out all night, night after night, and been seen each morning crying at dawn. We also learn that he is heartbroken over Rosaline's rejection, and when his cousin Benvolio begs him to act more sensibly and forget about Rosaline, he refuses. This reaction over Rosaline is certainly an intense, irrational, emotional reaction. It is these sorts of emotions that entice him to act hastily in marrying Juliet and also drive him to commit suicide, leading to his downfall.