SNK games: In a lot of the SNK fighting games, blood shed has been removed and some of the more violent moves are removed.
Bionic Commando (NES): In the Japanese version, you're fighting a group of Neo-Nazis. Naturally, there are swastikas everywhere. In the U.S. version, it's just "some terrorist group" with an eagle as their emblem. Funny, they took the "offensive" swastikas out, but they left the images of Hitler getting his face shot off in.

Just a few more edits in that game.
Puyo Puyo: More of an alteration. When two versions of this puzzle game where ported to the states, game developers figured that consumers would be turned off by the anime looking characters. So, more recognizable characters replaced the obscure ones and those games became Kirby's Avalanche and Dr. Robotnik's Mean Bean Machine.
If anyone else can think of anymore edits (name changes don't really count in my book), please contribute. Additions to U.S. versions (extra FMV in Final Fantasy VII) are also accepted.