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“”CORPORATION, n. An ingenious device for obtaining individual profit without individual responsibility.
|—Ambrose Bierce, The Devil's Dictionary (1911)|
A corporation is an artificial entity formed by one or more interested parties to achieve some specific end.
Usually the connotation is of profit-motivated, publicly-owned economic players, however the structure is also used to organize not-for-profit do-gooder projects, and some government functions are set up as semi-independent corporations.
 Who runs 'em?
A corporation (of the non-governmental for-profit variety) is owned by its shareholders. These shareholders may be individuals, other corporations, or -- and this is the funny part -- the corporation itself. Shares owned by the corporation that issued them are sometimes called "treasury stock," whereas shares owned by entities other than the issuing corporation are said to be "outstanding." Even more amusingly, under the laws of most corporate-friendly states, a corporation can just up and decide to issue brand new shares at any time, thus diluting the ownership interest represented by each of its existing shares.
Besides an ownership interest, shareholders of outstanding common stock get to vote in stockholders' meetings, their voting power being in proportion to the number of shares they own. There's not a whole lot these votes get to decide, but the few things they do get to decide are pretty major for the future of the corporation. Most notably, the shareholders elect a board of directors. These are the people who run the corporation from on high, and who hire its president, secretary, treasurer, and (if the corporation has one) the Chief Executive Officer (CEO).
Corporate shares can't be sold to just anybody. You can only buy stock in a corporation if a) you have some personal connection to the corporation (e.g. you're one of its officers or directors), or b) the corporation passes muster to have its shares publicly traded. Only those corporations that attain publicly-traded status can have their shares listed on one of the stock exchanges. Nearly all of the "big" corporations you've heard of -- Microsoft, Wal-Mart, Enron, etc. -- are publicly traded companies.
The assets of the corporation are separate and distinct from any of the people who run it. Corporations are free to open bank accounts, own property, sue, and be sued in their own names. If the corporation borrows money and then goes bankrupt, its stockholders, directors and officers are (usually) not liable to cover its debts; in other words, the corporate form of business ownership limits the liability of its owners and managers. For this and many other purposes, as a fiction of legal convenience, a corporation is treated as though it were a separate "artificial person."
 Are they people?
 In the United States
In the United States, a series of Supreme Court decisions has given corporations effectively all the rights of human beings, except voting in public elections. Holding these fictitious persons accountable, e.g. for actions that would get an actual individual tossed in the slammer for a lengthy term, has proven to be unrealistic.
Meanwhile, Congress has expanded the ability of corporations to declare bankruptcy, while restricting the corresponding rights for flesh-and-blood people. For example, student loans, auto loans, and credit card loans are all less discharged than corporate bonds, loans, and pension obligations to workers who have loyally worked at the same corporation for their whole lives.
Some argue taxing corporations while denying the right to vote in elections is taxation without representation. Nevertheless, corporate income tax rates in the U.S. are much lower than individual income tax rates. Supposedly, since the rich people who benefit from stock dividends and high executive salaries are taxed as individuals, this all evens out; but a corporation can do lots of tax tricks like giving its executives unlimited use of a company car and counting it as a business expense, without counting it toward the executives' taxable compensation.
 In the Commonwealth system
Throughout the Commonwealth, corporations are generally considered to be "legal persons" by the common law system for the purposes of statutory interpretation (i.e. figuring out what a law means). This means that, if any given piece of legislation says "a person," that is taken to include both "legal persons" (corporations) and "natural persons" (homo sapiens). This means that corporations have many of the same rights and can commit many of the same crimes as natural persons, although there are many cases where a corporation would not be charged with a crime (it is, for example, very difficult for a corporation to commit assault).