Saint-Simonians were members of a political and social movement that was often referred to as a utopian socialist movement. They followed the ideas of the Comte de Saint-Simon.
Saint-Simon did not believe in doing away with industrialism like some socialists of his time (1760-1825). Instead, he believed that entrepreneurs and workers would naturally come to see that they had a lot of interests in common. He believed that they would realize that private property was actually bad for efficient economic growth. Therefore, he believed, property should be owned by the state and everyone would become equal.
Saint-Simon also held other relatively radical beliefs. Notably, he believed that women should have equal rights to men. He believed that they, along with criminals and poor people, had been exploited by the capitalist system.
Saint-Simonians, then, believed in a socialist utopia in which all people would be equal and would be able to live harmoniously since they would have realized that their interests were compatible with one another.