Market: A system in which commodities are exchanged between producers and buyers (individuals and enterprises). When a large enough market emerges, commodities start to take on market prices - two sheep is worth one bushel of corn, etc. And eventually these prices are all related to a currency - one sheep is worth five gold, one bushel of corn is worth ten gold, etc. Generally an enterprise that does well in the market will persist, fluorish, etc., and those that do not do well will "go out of business", so to speak.
Enterprise: This isn't the most precise definition, but more or less an enterprise is an entity which buys components/tools/etc and produces some set of other products. Generally, an enterprise can be thought of as having two essential elements:
Socialism (Mutualist Definition)*: A socialist economy is when enterprises are run democratically by the workers, and the workers own where they work. Strict equality of payoff and ownership is not necessary, although any uneven-ness is likely a result of different labor input into the enterprise.
*This is the oldest definition of Socialism (this definition can be expanded upon, but the core features remain, therefore why it's a good starting point for understanding).
Capitalism: Someone with sufficient money could just buy the necessary Capital, and own it all. They could then hire people to work the Capital and produce the Capital, pay them a wage, and then keep the difference between labor+commodity costs and net revenue - profit. The Capitalist (unless under an explicitly socialist legal system) is then a dictator of this enterprise, and is free to sell all or parts of the ownership to others - workers have no say (unless they buy some of the enterprise, but since they are paid wages, this is very unlikely... unless government regulation or union efforts force higher wages...), as they have no ownership...
Unions: Except workers CAN have a say. If the workers unionize and strike, the Capitalist can't extract profit from the enterprise and its laborers, because nothing is produced. So unions are a way of forcing worker power, even when they don't have legal ownership over the enterprise.
Capital markets: This is when a group of people (typically, or about to become, the ruling class) have a bunch of wealth, and that wealth can be invested in fractional owernship of companies ("stocks"). As a result, this group of people - the Capitalists - get profit from what they earn, giving them more money, allowing them to buy more of the economy, garnering them increasing profits, and it's a self-reinforcing cycle. Capital markets allow the centralization of the economy into the hands of the few.
Fascism: Umberto Eco's 14 points on Fascism is a pretty popular set of criteria, but to get straight to the point, what is Fascism? As "Behind the Bastards" host Robert Evans puts it (paraphrasing), "fascists see the state as a gun, just another tool". Fascists harness popular discontent - driven largely by factors of Capitalism (and imperialism and its consequences) - into narratives that some "Other" has actually conspired to generate all of these problems, and to undermine the "traditional character" of a people. All of the problems of society can be explained as a conspiracy of the Other. This Other can be Jews, it can liberals, it can be socialists, or all of the above. Rather than pointing to the structural reasons why society is going bad, Fascists will elude towards conspiratorial explanations as the prime explanations of social woes. As such, only a trusted cadre, lead by a "morally righteous figure", can return "the nation (and by this, usually a specific ethnic group of said nation)" to its fantasized golden age. Because you can't trust anyone else - they're either in on it, they're a sheeple, or they're spineless/morally degenerate. And because this supposed conspiracy has so thoroughly compromised the nation, defeating the conspiracy SHOULD be achieved by any means necessary.
Expanded Socialism Definition: To more accurately deal with the woes of Capitalism and imperialism, the people at large can collectivize sectors of the economy, and officials are elected to manage them. In this situation, a public body becomes the primary Capital provider (although "private" means of Capital allocation are not out of the picture, they would just mean things like corporate bonds, rather than stock ownership, for private Capital raising), and different sectors could be run differently. Some industries are more efficient in a market setting (see the initial Socialist definition), and others in a planned setting.
(Socialists often disagree to what extent the market should be included.)
Here, the workers at collectivized institutions indirectly "own where they work" via their being part of the people. This is nominally what public goods are today - for example, highways, the police, and so on.
What Do I Want, or "Government corruption?" Like many things, dealing with corruption has A LOT to do with the road you take. I argue for a strong labor sector, and the democratic labor union road towards mixed socialism, in large part with this issue in mind.
Now you might imagine an alternative scenario - what if we had really strong labor unions, and kept Capitalists? If the main check on government in this scenario is the power to strike (and representative democracy is just a sweet cherry on top), what's the point in going Socialist? What's the difference - we still have concentrated power in either case!
The main difference is in a Socialist system, the government's centralized power is not self-reinforcing. In a Capitalist system, centralized power IS self-reinforcing. It can grow like a cancer, evading the shackles of regulation and the laws against union-busting. It can CORRUPT government, bending it to bust labor (this is what happened in most of the Western nations after WWII, and after 1980 - we've seen this strategy fail before). Furthermore, the representative democratic element is important - not only can a strong labor movement sink the economy, they can also vote for a different representative. They can pick someone else to be at the bargaining table. Is abuse IMPOSSIBLE? No, but in this situation the capacity for corruption and abuse is much attenuated.