There are many socialisms. The weakest form that many socialists (such as myself) don't consider socialism is social democracy, which accepts the market as legitimate and attempts to "fix it" or otherwise tame it with regulations and social spending. Obviously, this doesn't work because eventually, the ruling class decides it wants more money and undoes the regulations - the story of America since the last Gilded Age.
The socialisms to the left of social democracy focus on fundamentally changing the basis of the economic and therefore social system. These interventions focus on workers owning the means of production. In "market socialism" this means the market is left intact, but the old capitalist class is eliminated or otherwise largely disempowered in favor of worker ownership of private companies. If we go further left, we get state capitalism, where a democratic state owns the means of production and uses it to serve the people (with various levels of employee empowerment in the workplace). Communism is supposed to be a state where the state, no longer needing to suppress class struggle, fades away as being unnecessary. Many of these ideas are up for interpretation. I currently think that at least large parts of the security state would fade away, but am unsure about other public services and see the need for central authorities to coordinate on environmental issues now that we are facing catastrophe.
In the early part of the 20th century, "socialists" and "communists" distinguished themselves from each other. "Socialists" claimed the mantle of an electoral road to one of these outcomes, while "communists" focused on militant revolution, seeing the bourgeois state as hopelessly captured by the ruling classes / reactionaries / counter-revolutionaries.
Anywho, I hope that helps defines terms for y'all.