Can you please help me understand the difference between anarchism, communism, and marxism?
Very simplified version with probably lots of flaws:
Socialism is a system where the means of production are owned by the people and are used to meet the needs of all, as opposed to capitalism where the means of production are owned by a small elite and used to generate profit for that elite. The goal of Socialism is to lead to Communism; a classless, moneyless, stateless, egalitarian society.
The (main) difference between Communists and Anarchists is how they want to achieve that goal. Communists want to use the state as a tool to transition from Capitalism to Socialism to Communism, at which point the state would no longer be needed. Anarchists
oppose authority and argue that you cannot create equality using hierarchical systems, and so want to do away with the state immediately.
Marxism describes the Economic and Philosophical theories of Karl Marx. Among other things he described how society progresses from one type of society to another through class struggle, and described how the logical inconsistencies inherent in Capitalist society must inevitably lead to its destruction.
Some good introductory books: