Extreme simplification:

Liberalism: supports capitalism. Current system + tweaks

Leftism: supports anticapitalism of some form, the two biggest umbrellas being Marxism/Communism and Anarchism

Marxism/Communism: supports collectivization, public ownership, and central planning (I have an introductory reading list if you want to learn more, or just read Principles of Communism)

Anarchism: supports full horizontalism and networks of communes