OP, if you take nothing else away from this conversation, it is that different people have different notions of what exactly the word "socialism" refers to, which in practice makes it a useless word to use in the context of discussing public policy because you just end up with groups talking past each other. In the most extreme case, if someone thinks you are proposing "socialism", then they might abruptly stop listening to what you are actually saying and assume that what you are actually proposing is to turn over the entire country to a corrupt authoritarian government because that is what the word "socialism" means to them. For this reason, should you find yourself in a discussion about public policy, it is generally better to be very specific about exactly what policies you are saying are good or bad and why you think they are good or bad without resorting to using what are in practice ambiguous and loaded terms like these. (Just to be clear, I am not saying that this state of affairs is reasonable, just that this is how it is at the moment.)

In real general terms, communism is about people/state’s ownership of the means of production. Under this system, most private property is nationalized.

Socialism allows for private property and sees the role of the state to redistribute power and wealth among its citizens through some sort of state program.

It depends a lot on what you actually mean by socialism and communism because these words can have very different meanings to different people and ideology.

As a very broad baseline, socialism is the socialization of the means of production, as opposed to the current privatization of those means. Now there are a lot of ways this could be done, and thus a lot of ways to define socialism. Some socialists want a strong State that can enforce strict rules of ownership, others want no State at all and a free cooperation between individuals, with a lot of variations in between. An anarchist, a communist, a social-democrat would all consider themselves socialists, even when they actually have very distinct ideologies.

Now communism, at least in its most recognizable form, is basically the end state of socialism in the Marxist ideology specifically. It designates a stateless, classless society in which each person contributes according to their ability and receives according to their needs. It's basically the end goal theorized by Marx that has never been achieved yet in History.

Communism is socialism, but socialism is not necessarily communism. Socialism is the counter part to capitalism, and communism is a form of socialism. I found this video very helpful when I had the same question.

Lots of people has great answers here. I would love to explain a bit of the reasoning behind these production relations in very crude and hand-wavy terms.

At the time of Marx, people see the economy consists of two major inputs, labour and means of production (land, machinary, tools, etc), with machinary and tools (things that can be produced and enables production) called capitial.

In a capitalistic society, like its name suggest, capital is a valuable resource. That means people owning these capital, aka capitalist, can make money just by renting these capitals without contributing labor.

However with the productivity increase, it was theorized that capitals will lose its value, hence enabling workers to collectively own the means of production. At that stage, only labour will be valuable resource, hence the compensation will be directly tied to the value of labour a worker can provide.

Finally, when the productivity is way over the capacity of consumption, communism is achieved. Human will no longer fight for resource, since the amount of resource can fullfill the need of any individual. Thus the society will be able to distribute resource simply by need.

However like other has said, there are many means of achieving these ideals, not necessarily by pure market changes or by a authoritarian state.

A lot of words but not many sources here. So here's a few:

Marx defined socialism as: "...Socialized man, the associated producers, regulate their interchange with nature rationally, bring it under their common control, instead of being ruled by it as by some blind power; they accomplish their task with the least expenditure of energy and under conditions most adequate to their human nature and most worthy of it. But it always remains a realm of necessity. Beyond it begins that development of human power, which is its own end, the true realm of freedom, which, however, can flourish only upon that realm of necessity as its basis."

— Capital III, translated by Ernest Untermann, Charles H. Kerr & Co., Chicago 1909, p. 954

To understand Marx's definitions you have to realise he was writing mostly in response to the Paris Commune uprising and therefore saw 'communism' as the practical application of a theory of socialism. However, the terms and their meaning were radically reshaped by Lenin, Mao and Stalin.

— The Paris Commune: First Proletarian Dictatorship, Revolution, Vol. 3, No. 6, March 1978.

In March 1918 the Bolshevik Party was renamed the Russian Communist Party (Bolshevik) in order to distinguish it from Social Democratic parties in Russia and Europe and to separate the followers of Lenin from those affiliated with the nonrevolutionary Socialist International.

— https://www.britannica.com/place/Soviet-Union/Lenin-and-the-Bolsheviks

I think this video helps

https://youtu.be/lrBRV3WK2x4