I am going to copy and paste the definition of Anarchism I posted in another thread. It apparently helped a lot, so, here it is again.
Anarchism is a society with no definite hierarchy. That means getting rid of the capitalist class structure so one person can't have economic power over another. It also means getting rid of political hierarchy, so no one has political power over another.
That means all property is owned by the public, as with socialism. It also means that decisions are made by the workers and common people, not by the politicians and elected "representatives".
Instead of "government" as we know it, an anarchist society has a federation of autonomous communes, in which the people who live in each commune make the decisions democratically.
And as for the no authority part, you're mostly right. Anarchists are against authority, except for rational authority, which is authority that one doesn't have to listen to, and authority that is backed up by experience or knowledge, rather than by status or class. For example, when talking about what material to make a good pair of shoes out of, you would refer to the authority of a shoemaker.
Feel free to PM me with questions on Anarchism.
EDIT: Also, Anarchism is left-wing as it is socialist in practice. That is, no private property, just as with communism/socialism.
I'm on some sickle-hammer shit
Collective Bruce Banner shit
FKA: #FF0000, AKA Mistake Not My Current State Of Joshing Gentle Peevishness For The Awesome And Terrible Majesty Of The Towering Seas Of Ire That Are Themselves The Milquetoast Shallows Fringing My Vast Oceans Of Wrath