Take a look at this, gives a fair summary...
Socialism = The State owns the means of production (rather than private companies), and distributes the result
Communism = No State/Government nor any private companies. The People themselves decides communally what they need and how to best use resources. Has never been done!
The idea was that Socialism would replace Capitalism, and that Communism would replace Socialism. But while Communism may initially have been the "final goal", "communist" states like the Soviet Union, remained stuck in Socialism (probably because "power corrupts, and absolute power corrupts absolutely").
Of course, many means "social-democracy" when they talk about "socialism"; I.e. a system where there are private companies, but where the Government provide lots of services (like school and healthcare) and perhaps owns companies for key infrastructure (e.g. telecommunication, train&rails, and radio/TV).
Marxism was Carl Marx's theories about communism and how it could be achieved... as opposed to Leninism, Stalinism and Maoism - which all may have had more or less the same goal, but differed in how to get there.