What it is: This is a Greek word meaning 'no rule' or 'no government'. It is commonly used to describe lawless and disorderly behaviour. Anarchists themselves advocate the replacement of the state by voluntary co-operation.
What it means: Anarchy can lead to 'chaos', which means complete disorder and utter confusion. In politics, anarchy is rather less threatening. Political anarchists think that there should be no governing power. Instead people should live in a 'free society' in which order is kept because people co-operate with each other in sticking to an agreed set of reasonable rules. Anarchist societies have been tried out from time to time throughout most of human history, but seem to have worked only when relatively small or kept together by a strong shared belief in something else (religion or cultural customs, maybe) as well as political anarchy.
Think about it: Political anarchists are critical of the whole idea of people being organised in nation states. Maybe it's worth at least trying to imagine some other ways of living in the world. Could a 'free society' with agreed rules work on a large scale? If so, how?
For a wider discussion see Resisting the Nation State