At it's most basic, a corporation in Eve is a formal group of pilots. Like the real world, the corporation is run by a single CEO, who may or may not appoint Directors. Directors are, for all intents and purposes, the same as a CEO, but would report to the CEO.

Depending on the size of the corporation, roles may be created to manage specific areas of the corporation. As the corp grows, the demands on the CEO can grow to the point that he is managing the corp and not really playing the game. That's where these formal roles come into play (and yes there are actual game mechanics to these roles, for example a recruiter can accept applictions but may not be able to access corporation assets, just depending on the amount of access he has been given).

Common Roles:

Fleet Commander
Field Commander
Mining Foreman
Manufacturing Foreman
Recruiter

Other roles may be created with custom mechanics to manage parts of the corporation.

Pilots must apply to join a corporation, and there is often an intensive interview process.

Corporations can also declare war against other corporation. This is a mechanic that allows a shooting war even in space where PvP is not generally allowed (high security). As there are thousands of corporations, this means that there is never a shortage of war targets, if that is your thing. By comparison, in WoW, if you are human, you can only attack Horde. Someone on your side you don't like? Too bad. In Eve, someone you don't like is in your corporation, you can kick him (if you have those roles) or you can leave and form a new corporation. You can then declare war on his corporation (provided he isn't in an NPC corporation).

Additionally, corporations can set official standings toward other corporations. A "blue" corp is friendly; a "red" corp is not. While these have no mechanics other than flagging those pilots to you, they allow you to know friend from foe in non-shooting situations (though in low security space you may still decide to attack them. These standings open up the door for true diplomatic relations between corporations; as a Director of an alliance, I know all about these. "Blue" status is very valuable in this universe.

Corporations also own assets, like Player Owned Structures, or POS. These are not space stations in the classic sense, but do provide a safe haven for pilots and other individual assets. They also provide the means for industry not otherwise available. (Industry equals alchemy in other games.)

Corporations have their own goals in the game (a PvP corp, a PvE corp, an industrial corp, etc), but may join up officially with other corporations to form Alliances. These alliances work together, often with the goal of claiming sovereinty over solar system, and entire regions of "null security" space (no protection from the in game "police").

Hope this makes sense.

It doesn't matter because clans can't fight corporations or behalf of other corporations (unless they allow you to go solo and accept contracts from a corporation without the need of your own corporation being assigned one), and why would you spy on a clan anyway?

Not really because, well... they really can't do anything.

It doesn't matter if a clan has spies or not, there's really nothing to spy on.

Well people (like myself) might feel a little touchy when EVE people say this: