According to Wookieepedia, in the original 1977 Star Wars film (now known as A New Hope), after the bounty hunter Greedo says "I've been looking forward to this moment for a long time" (talking about killing Han Solo in Mos Eisley cantina), Han shoots Greedo under the table while Greedo was pointing a blaster at him.
In the remastered 1997 Special Edition, George Lucas changed the scene so that Greedo shoots at Han and misses, without explanation, from point blank range, and then Han shoots him in self-defense. Later, it was altered again to have Han and Greedo shooting at almost the same time (though Greedo still shoots first), with Greedo still missing from point blank range.
"The principal objection from critics seems to be that the change dilutes and compromises Han's rebellious and ruthless nature. The change is felt to detract from Han's 'anti heroic' qualities, and also diminishes the character's growth and development over the story from a Machiavellian smuggler who cares only about himself (and his co-pilot Chewbacca) into a committed member of the Rebel Alliance fighting to bring freedom to the galaxy, as his shot is clearly in self-defense."
You can read a lot more discussion of the controversy at "Han shot first" on Wookieepedia.