Here's a good short explanation.
In very broad strokes, socialism is an economic system in which "the means of producing and distributing goods is owned collectively or by a centralized government that often plans and controls the economy." In modern societies, socialism often attempts to eradicate class divisions. While the word "socialism" is sometimes used interchangeably with "communism," the two aren't the same -- communism is a more extreme form of socialism.
Communism advocates the "collective ownership of property and the organization of labor for the common advantage of all members." While communism is first and foremost an economic system, it's also a political ideology that rejects religion. And just as communism is a form of socialism, Marxism, Maoism, and Leninism are branches of communism.
Like socialism and communism, fascism uses a central authority to maintain control, but "terror and censorship" are common. It results from economic failure in democratic political systems. Interestingly, while socialism and communism are both on the left end of the political spectrum, fascism contains elements of both "left and right ideology" and rises from economic collapse. The most famous fascist was Italian dictator Benito Mussolini. That ought to tell you it's not a good way to run a country.
