You already know the story and it's an old meme, but let me sum up anyway. In the original Star Wars, Han Solo shot Greedo the bounty hunter in the Cantina - in cold blood. In the first revised edition, George Lucas changed it so that Greedo shot first (and missed) and Han shot him back. In the second revised edition, Han and Greedo shot at the same time. Greedo missed. Han didn't.
...and an entire planet of nerds were outraged.
Lucas said he made the change because Han was a hero and heroes don't shoot first. Far be it from me to tell him he was wrong to make that change, but since this is the internet, I'm going to do it anyway.
Han shooting first improves the story in two different ways. Firstly, if you can, think back to the very first time you saw Star Wars - assuming you are old enough to have seen the original version. Luke and Ben go into this seedy bar to hire a pilot to take them to some other planet. After they hire the guy, we see him calmly murder someone, admittedly someone who looked like he deserved it. This immediately jacks up the tension for the audience. Oh Crap! The pilot they hired is a murderer! Surely, he'll turn on them before they get where they are going.
This doesn't happen, of course, but the first time you see the movie, you don't know that. This tension is lost if Han doesn't shoot first.
Secondly, let's look at the long term character development of Han. Through the course of two movies, he evolves into a self-less hero - so selfless that he gives himself up to Boba Fett in order to save his friends. Him being lowered into the carbonite chamber is the climax of his development. But it is where he evolves from that has been changed. In the original, Han starts as a cold blooded murderer and grows into a selfless, though roguish hero. In the updated version, Han merely starts as a small time crook trying to project the image of being a badass.
I think this is one place where Lucas got it right the first time.