In the original version and even in the rough cut of this scene, Han always shot first in order to defend himself from being killed right then and there. Greedo clearly tells Han that he's about to do it.

Rough cut dialogue -

Greedo: Jabba would rather have your ship.

Han: Over my dead body.

Greedo: (laughs) That is the idea, Solo. Now come outside with me... or must I finish it here?

Han: I don't think they'd like another killing in here.

Greedo: They'd hardly notice. Get up! ("I've been looking forward to this fior a long time" is not heard, but there is enough silence for this to have been added later.)

Han: Yes. I bet you have.

BANG!

Original Theatrical Version dialogue -

Greedo: You can tell Jabba. He may only take your ship.

Han: Over my dead body.

Greedo: That's the idea. I've been looking forward to this for a long time.

Han: Yes. I bet you have.

BANG!