The way I have heard it described is that governments have a monopoly on the "legitimate use of force." That is, if someone commits a crime, or doesn't pay taxes, or doesn't hold up their end of a contract, the government is the only entity allowed to physically (or otherwise) force someone to do something (go to jail, pay a fine, etc.).

I like this definition better because it avoids some of the objections people have responded with. 1) it avoids the word "violence" - the use of force does not require violence, and I think most people would hope that governments wouldn't use violence in their enforcement of laws (though we know that in practice this is often not the case), though the threat of force is more consistently necessary) and 2) it acknowledges that it's not a monopoly on the use of force, as the reality is that many people use force, but on its legitimate use.

I know it's even called "monopoly on violence" in the wikipedia article, so it's not that I think you're mistaken, but rather I prefer this definition. The article mentions the term "monopoly on violence" in English is indeed common, but also controversial.

http://en.wikipedia.org/wiki/Monopoly_on_violence