Prior to the tragedy, the Wayne family were walking home from a film based upon the character known as Robin Hood. Young Bruce told his dad that he thought that "Robin Hood was something else." They take a short cut through Park Row, and that is where they are confronted by a mysterious mugger named Joe Chill.
Joe Chill confronts the family and attempts to rob them, trying to take Martha Wayne's purse, but he is met with resistance by Thomas Wayne and so he pulls out a gun and shoots both of them, and Bruce witnesses the murder. They drop to the ground, under a rainy Gotham City sky. Bruce sobs as he watches his parents bleed to death.
Chill, realizing he had killed both of the young boy's parents, panics and runs away. He dropped the purse he had attempted to steal from his victim, instead abandoning it. What this tells us about Joe's character is that he was likely appalled by what he had done, probably not meaning to go that far.
It also could indicate that he was just afraid that someone would see him with a purse that was clearly not a man's handbag, and it could draw attention to him, especially after gunshots were fired. Still, this is likely not the case, as the gunshots were fairly muffled by thunder striking nearby.
The Park Row tragedy was what led Bruce Wayne to choose a career as a crimefighter. He went through a number of different types of training, both physical and mental. With the help of his guardian Alfred Pennyworth, he would develop the Batsuit, and a number of other gadgets and vehicles to help him fight crime as Batman.
As for Park Row, it from then on became known as Crime Alley, because of the terrible tragedy that shattered the Wayne family. Bruce Wayne himself was traumatized by the alley, and he never returned there again, because of his fear of it. He never returned that is, until he was able to conquer his fear. This wasn't until 1985. At that point, every year, on the anniversary of his parent's deaths, Bruce brings roses to this alley and places them in memory of the event and people that made him what he is.