Opposition to hierarchy.
From Greek for without rulers. Anarchism is rooted in opposition to forms of domination and coercion; historically, its three main opponents have been the Church, the State, and capitalism. The main Western tradition of anarchism dates back to the advent of socialism, although there have been many anarchic societies and philosophers the world over for pretty much as long as people have existed.
There are lots of forms of anarchism, generally relating to either different visions of an ideal horizontal social structure, or just boiling down to what that particular activist focuses on (anarcho-feminists are mostly concerned with patriarchy, for instance). Mostly, anarchists aren’t prescriptivist - as long as people are trying to take power into their own lives, anarchists are for it.
No, because the plumber doesn’t have power over you.
Your boss has power over you - he can decide to remove your source of income, and uses that as leverage to get you to do what he wants.
Your government has power over you - they have a monopoly on the use of violence and use it to maintain control.
Your preacher has power over you - by making you afraid for your soul, he can manipulate you unwillingly, or severely impact your social status.
Your plumber is someone providing a service, that’s completely different.