Anarchism is a sub-ideology of communism that seeks to abolish all kind of unjustified hierarchy, including capitalism. It's the farthest left ideology.

"Anarcho-capitalist" is a strange concept that's very recent and only seem to exist in the USA, that tries to reimagine feudalism in the industrial age, meaning that very rich people are free to have their own army and own massive area of land, where people living there will subjected to the laws of that lord, with no possibility to remove them through voting. It really doesn't have anything in common with anarchy.

You can read more on the wikipedia page, it seems to be pretty good.