This is actually quite a good question, and I'm sure there is a textbook somewhere that describes the state, but from a libertarian perspective, I'd say that it is any grouping or section of society that seeks to monopolise violence. Everything else (wars, "justice", taxation) that the state does, stems from their hold on violence.
If you think about the power of the state, it stems from the fact that we all think violence is icky and don't want to sully our hands with it. And it's true really, nobody sane wants to confront a burglar with a baseball bat or a shotgun. So having a group in society that is prepared to take this unpleasant chore from us sounds like a great deal.
And to keep us sweet, the state also offers us "services" like gimmicky health care and "education", specifically education that is tailored to making the state seem like a good idea.
The problem with the state is that once it has obtained that monopoly in violence and inculcated us with a belief that the state is the only proper wielder of violence, it is then quite easy and natural for the state to increasingly turn that violence on us. For example, you notice how "crimes" against the state get punished strangely harshly, while repeated crimes against individuals or property get a ludicrous slap on the wrist ... unless those crimes threaten the violence monopoly of the state.
One of the great burdens of responsibility that comes with the freedoms of anarchy is the responsibility to defend yourself, to protect yourself against crime and to exercise violence when it is needed. But as a wise man once said, "with great power comes great responsibility". And the power to live your own life to its very fullest must surely be the greatest power of all.
This may not be the definition you might expect of the state, but from an anarchist position, I think it's a valid one.