Han shoots second: what implications does this have for the character of Han Solo?
My answer is: None whatsoever. This change, made for the Special Edition of Star Wars in 1997, does nothing to affect Han's characterization, George Lucas's claims to the contrary. Apparently Lucas decided that Han wouldn't actually shoot Greedo in what appears to be cold-blood, so he altered the film so that Greedo shoots first, and then Han shoots in return. The general criticism of this scene is that this change totally changes Han's character, blunting his rogueish nature and making him "softer". I don't think it does this at all, really.
What do we learn of Han Solo in this scene? We learn that he owes money to Jabba the Hutt, we learn that Jabba has enforcers out looking to collect the debt, we learn that Han is pretty cool in such a crisis, we learn that Han's general approach to such problems is to blast his way out of them, and we learn that Han is a very good shot with the blaster, a weapon that Obi Wan Kenobi has already characterized as "clumsy" and "random". I'd say that in the Special Edition version of this scene, we still learn all of that.
As for Han not being one to shoot first, well, Lucas is flat-out wrong. (And I'm a steadfast defender of Lucas, so for me to say he's wrong is quite a thing.) Consider that in just about each tight spot in the films, Han's initial preference is to fight. He wants to fight his way out when the Millennium Falcon is caught by the tractor beam; he wants to fight when Luke complains that Chewbacca is howling too loudly; when a cadre of stormtroopers comes across the heroes in a Death Star hallway, Han blasts the guy in front and charges after them. In The Empire Strikes Back, Han's blaster is in his hand instantly when the door opens on Cloud City to reveal Vader. So on and so forth. Han's more than willing to shoot first if he feels the need to do so, and he feels that need quite a bit over the course of the films. In fact, I'd say that his reliance on subterfuge on Endor in Return of the Jedi constitutes an overlooked bit of character growth on Han's part. (Many fans think that Han is just "along for the ride" in ROTJ, but he has a very real character arc that plays out fairly subtly.)
The fact is this: if the altered Greedo scene really hurts Han's character, then we can expect that Han's subsequent actions would not be consistent with that scene. I don't think that this claim can plausibly be made. He's cool under pressure, he gets his weapon at the ready, and when the moment comes, he pulls the trigger without hesitation. So I don't think that having Greedo shoot first harms the characterization of Han Solo at all.
What Greedo shooting first hurts, then, is actually the scene itself. In the original, it's a very funny scene; here it's less so, because the scene has a different flow to it (better in the DVD version, which was altered again to make it work better than it did in 1997), and because as the scene exists, Greedo has no reason at all to shoot when he does. There's no logic to Greedo shooting at all, unless Han makes some kind of move that startles him, or if Han says something that articularly enrages Greedo, or so on. As it is, the scene goes from being a funny scene that also establishes Han's character to being a scene that is less funny because, although it still establishes Han's character, it makes less sense. It's the equivalent of a joke that is told masterfully until the punchline, which the joke-teller muffs.
The other major addition to this part of Star Wars is the scene between Han and Jabba himself, originally filmed with a stand-in actor playing Jabba but never finished because the effects couldn't be done. (Incidentally, Jabba wasn't even designed until his appearance in Return of the Jedi, so presumably the Jabba originally intended for A New Hope would have been very different from what eventually showed up.) I don't object that the scene repeats quite a bit of the dialogue in the Greedo scene. The final line, though -- Han saying "Jabba, you're a wonderful human being" -- is just horrible. In the original script, the line was "Jabba, I'll pay you because it's my pleasure". That would have worked a lot better. As it is now, I'm wondering why Han is calling this big-ass slug thing a "wonderful human being". Is he making fun of Jabba, because he's so fat and Han's thin and can walk around and stuff? What is he saying here?
Anyway, there's my answer to that question. Does anyone have any others? Post 'em in comments at the link above, or even here. Test me, folks! Challenge me! This is your chance to make me talk about something goofy (or something).