Capitalism is an economic system under which private individuals own the means of production. These "private individuals" may take the form of Corporations, individuals, or even partnerships. Fundamental elements of Capitalism are free market operation, competition and no (or at least limited) government intervention. Workers may be capitalists themselves, such as self employed Carpenters, professionals, etc. or they may be employed by those who do, such as factory workers. Under Socialism, the workers themselves own the means of production. Quite often this form of ownership is by the state itself; which is controlled by the workers. Under a socialist system, competition is practically non-existent; and markets are somewhat limited. A big drawback of Socialism is that lack of competition often results in inferior production and shortages.
Socialism was somewhat re-defined during the Communist Era; although that system was hardly a true Socialist one. In every instance, there was actually a state Dictatorship which forced a socialist system on the workers, largely against their will.