Why did he fall to the ground?

Ah, one of the great movie questions. I'm surprised it hasn't come up here before. This particular Mexican standoff scene has been dissected quite a bit, and the IMDB link provided by @GertArnold gives the consensus:

The bullets were supposed to fly thusly: Joe shoots Mr. Orange, Eddie shoots Mr. White, and Mr. White shoots Joe and then Eddie. However, during filming Eddie (Chris Penn)'s squib went off before Mr. White was able to get to him, and he fell to the floor anyway. Hence, the confusion began. Tarantino has claimed that he realized the mistake, but left it in the film to give people something to talk about.

The only other possibilities are either Mr. Pink or Mr. Orange, but Pink was hiding and Orange had just unloaded his gun into Mr. Blonde:

Seeing as how Mr. Orange did not have a loaded gun at that time, Mr. White is really the only possible candidate for Eddie's shooting and if you watch the scene in slow motion, as Mr. White is falling, he clearly turns and fires at Eddie.

(The scene can be watched in slow motion here on Youtube.)

Finally, an excerpt from the original script backs up that it was Mr. White:

Mr. White SHOOTS Joe twice in the face. Joe brings his
hands up to his face, screaming, and falls to the ground.

Eddie FIRES at Mr. White, HITTING him three times in the
chest.

Mr. White brings his gun around on Eddie and SHOOTS him.

While the shootout unfolds slightly differently in the script to what is on screen, it is mainly superficial details such as what is said. The characters' intentions are maintained, which clears Mr. Orange and Mr. Pink, who were present but not responsible:

Mr. Orange lies perfectly still, except for his chest
heaving.  The only SOUND we hear is his loud breathing.

Mr. White is SHOT full of holes, but still on his knees,
not moving.

Mr. Pink is standing motionless.  Finally he grabs the
satchel of diamonds and RUNS out the door.

TL;DR:

It was Mr. White after he shot Joe.

Though some argue that Han shot first[citation needed].

Larry (Mr. White) shot Nice Guy Eddie.

It's all a bit unclear and very quick (the frames above are from about 0.5 seconds), but thankfully Chris Penn (the guy who plays Nice Guy Eddie) explained it:

"It was a mistake," Penn later explained. "What was supposed to happen - and I don't know if Quentin's gonna like me giving this away, but it's too late now, he never told me not to - was Harvey Keitel [Larry/Mr. White] was supposed to shoot Lawrence Tierney [Joe Cabot], then shoot me [Nice Guy Eddie], then get squibbed. But what happened was the squib [a small explosive charge resembling a bullet hit] on Harvey went right off after he shot Lawrence, so he went down, but my squib went off anyway, so I went down. So, basically nobody shot Nice Guy Eddie. Quentin said 'You know what? It'll be the biggest controversy of the film. We're leaving it.' He was definitely right… " (Source: 2006 Telegraph obituary of Chris Penn. )