1. An economic system where the means of production, distribution and exchange is determined by the state/public sector in some form. Can be centralised, decentralised, democratic or undemocratic.
2. Description of a left-wing political position between social democracy (general acceptance of the market economy but thinks the public sector has a vital role in proividing some goods and services) and communism (marxism). Agrees in the state determining the means of production, distribution and exchange but wants to bring that about peacefully and democratically.
3. General description of the left: the belief that individuals should be judged on how they treat other people rather than on their job/race/sexuality, that people should have equality of opportunity, that in principle wealth should be distributed fairly to everyone who works rather than the minority who own most of the economy and most of the wealth and that an economy owned by a few individuals without a strong public sector to balance that is undemocratic and unjust.
4. A stage in history defined by Marx's theories as coming after capitlalism and before communism where the means of production is owned by the state and run in the interests of the proletariat.
5. A label used by various Marxist-Leninist dictatorships with state-run economies in the 20th Century to justify their totalitarianism.
All marxists are socialists but not all socialists are marxists.