in terms of developmental stages as laid out by marx, socialism is the lower stage of communism, after a revolution has happened and the workers control the means of production, during the class-dictatorship of the proletariat, before the state has withered away. During this phrase there are necessarily some trappings of the old society.

But that's the developmental context of the word socialist. The word socialist has many other contexts, some good, some bad. The nazis called themselves "national socialists" but Marxist-Leninists and Democratic socialists don't consider them such, viewing them as fascism masquerading as socialism (something fascism is incidentally very good at, since it capitalizes on the fury of downwardly mobile petit-bourgeois by channeling their anger against scapegoats instead of capitalism, i.e. channeling their potentially revolutionary energy into counter-revolutionary energy.)

The original sense of the word "socialist" which predates Marx and Engels was Utopian Socialism, which was born out of the European enlightenment. It was an idealist, non-materialist tradition of viewing that a vaguely defined Socialism (the end of class exploitation) would come about purely out of the development of technology, and/or the altruism of the capitalist, and/or people simply convincing others that it was the right system through civil debate. This utopian socialism came from people like Owen (a capitalist who tried to be nicer to his workers than other capitalists) and Fourier, but further developed through people like Lassalle in the Gotha programme, which Marx criticized. Marx and Engels took Utopian Socialism and made it "scientific" (in the 19th century sense of the word) by arming it with the economic critique of capital, synthesizing it with historical materialism, and emphasizing the need for the proletariat to become class conscious, militant, and organized, and banishing the illusions that socialism would simply come about peacefully through reasoned debate or reformist transition.