It depends on the situation, really. Basically, how appropriate it feels in context. The violence in something like Saving Private Ryan doesn't bother me much, because war is pretty darned violent. The violence in Kill Bill is blatantly over-the-top and tongue-in-cheek, so it amused me more than anything else. Meanwhile, I hate gratuitous violence. In fact, I probably would have enjoyed the new King Kong at least a little bit were it not for all the gratuitous violence. Instead, I ended up hating it, and feeling vaguely ill for days afterwards as the unpleasant imagery from it kept returning to me. Same thing with the godsawful movie version of Starship Troopers. Even though it was supposed to be a war movie, the violence in it seemed far, far too graphic and forced.
Video games... it depends on context, again. In all honesty, though, I don't have much of an opinion on it, because I've never bothered playing any games with the kind of violence I hate. Stuff like Duke Nuke'm 3D... you're fighting off aliens who took over the Earth and stole our women, so how can you _not_ want to see them die horribly? ;P Stuff like Grand Theft Auto I just never bothered with. The only time I want to be the 'bad guy' is when the people I'm attacking aren't presumably innocent. If they've done something to truly earn the main character's wrath, it's typically okay by me.
I think, though, that my biggest hangup about violence is how much more readily it's tolerated than sexuality and nudity, which, IMO, should not be considered 'bad' so long as people are being responsible, and no one is getting hurt without consent. Our anti-sexuality/nudity sentiments are trained into us by our parents, teachers, and peers. Looking at it logically and sensibly, there's no valid reason for this. Meanwhile, we don't have it drilled into us nearly enough how important it is not to take pleasure in the pain of others, while in reality, voyeuristic sadism is extremely common and typically tolerated and seen as perfectly normal/acceptable.