The Greek prefix "a" literally applies the meaning "not". Contextually this can be taken to mean "no", "without", "against", or "in the absence of". The prefix "an" is derivatives of the prefix "a", with the "n" appended for phonetical reasons, the same as we say "an apple" instead of "a apple", because it rolls off the tongue more easily, sounding less stilted.
The literal translation of "archos" is "chief", which depending on the context can be interpreted as "boss", "ruler", "King", "commander" etc. What it doesn't literally translate to is "state" or "government": the word for that is "cracy".
The word anarchy isn't spelled "anarchos" though, which would literally mean "no chiefs", instead it's derived separately from the root "arch" meaning "structure. This is the same root used in words such as "hierARCHYy", "monARCH", and "oligARCHY". Contextually it means "an-ARCHy": "no command structure". Literally it means "no rulers".
Anarchy represents personal sovereignty and the autonomy of the individual. Anarchism is a political theory of how to eliminate command structures from society... Because that's what individual autonomy really is, the absence of an imposed or coercive authority structure. The absence of arbitrary hierarchy. A society organised horizontally instead of vertically, wherever possible and practical.