A corporation is a legal entity, with all the rights and powers of a person save one: corporations can't vote. People form corporations for a number of reasons, which include legal protection of the owners - you can sue a corporation, but generally you cannot sue the owners. Also, a corporation lets many owners pool their money and accumulate more money than an individual could.
Corporation ownership is divided up into shares. Generally, people can buy and sell shares. The corporation is owned by the current list of share holders. Shares are also called stock. The stock holders vote, typically once a year, to elect directors. The directors meet and make decisions for the company, and are responsible to hire and fire the president of the company.
At the end of the year, if the corporation has made money, the directors may choose to declare a dividend. This means some of the profits will be divided up equally among the shares of stock. For example, if the directors declare a dividend of $2.50 per share, and you own 6,000 shares, then you will get a dividend check for $15,000.
Sometimes a corporation will issue several different types of stock with different rights. Shares of stock which allow you to vote on who is a director are called common stock. Sometimes there will be several types of common stock. For example, Berkshire Hathaway has class-A and class-B common stock. Their Class-A common stock sells for 30 times as much money per share as their Class-B common stock. Each Class-A share of Berkshire Hathaway has 200 times as big a vote as each class-B share. The directors can make any rules they like about different classes of stock, prices, convertibility, and voting rights. If you own Berkshire Hathaway class-A shares, you may return them to the company and they will issue you 30 class-B shares for each class-A share you turn in. However, you may not convert class-B shares to class-A shares. So, the price of class-A shares cannot fall below 30 times the price of class-B shares - if it ever did, people would convert class-A shares to class-B shares, thereby increasing the supply of class-B shares and decreasing the supply of class-A shares. However, class-A shares could raise to a higher price, as class-B shares are not convertible.
Another type of stock is called preferred stock. Preferred stock has no voting rights, but is usually guaranteed a dividend each year. Preferred stock used to be sometimes called "widow's stock" as this was thought to be a good way to take care of your wife after you died. The directors may, for example, decide to issue preferred stock at $100 per share with a guaranteed dividend of $6 per year. Buying this kind of stock is therefore a lot like putting money into a bank account, except the dividend represents an interest rate which is usually higher than banks pay. This is a way for corporations to raise money without a loan, which must be repaid someday. Because of the guaranteed dividend, preferred stock usually has a price which does not fluctuate very much, so long as people believe the company is not close to bankruptcy. In a recession the preferred stock may have its price go up a bit, reflecting that people want less risk in these times. If the economy is doing very well, the price of the preferred stock may go down a bit as people sell it and buy common shares, hoping that the price of the common shares will raise substantially as the company's profits increase.
The stock market is a place where people gather to buy and sell shares of stock in various corporations. The price of particular shares goes up and down minute by minute depending on how many people want to sell, how many people want to buy, news about the economy, and news about the particular company. In order to buy or sell shares on the market, you must be a member of the market. So for ordinary people to buy and sell shares you must find a broker who is a member of the stock exchange, and pay him a fee to buy or sell the stocks for you.
Some corporations issue stock options to valued employees. This means that on a particular date, the employees are allowed to buy a particular number of shares of stock from the corporation at a particular price. If on that date, the option date, the company's shares are trading for a higher price, then employees will exercise their option and buy the shares at the reduced price. If on the option date the market price for the shares is lower than the option price, then the option is said to be worthless, as you could buy the shares cheaper on the exchange than from the company. Stock options are a popular way to give a bonus to upper management, as then the upper management is well motivated to make the stock price go up.
If you buy up 51% of the shares in a corporation, then at the next election you could elect yourself and your friends as directors of the corporation, and then you could make anyone you wanted be the president of the company. If you take the total number of shares in a corporation and multiply that by the price per share, you get the price you would have to pay right now to buy the entire corporation. This price is called the market capitalization, or market cap. If you divide the market cap by the corporation's profits, you get the price / earnings ratio. Generally it is thought that stock should trade with a PE ratio of 10 to 25. During the Internet stock bubble, many Internet company stocks were trading at PE ratios of 200, 400, even 600 to one. In fact, many of these corporations had no earnings at all, so they didn't actually even have a PE ratio. In 1999, Internet companies would issue new shares of stock, and trade those shares for shares of other companies. This was compared to "printing money." It was in exactly this fashion that AOL bought Time-Warner, a far bigger and far more profitable company. However, AOL had a far higher share price, so although there was no way in the world AOL could have raised the cash to buy Time-Warner, they found it quite easy to issue stock and trade it for Time-Warner stock. At the height of the stock bubble, there were many Internet companies that were only a couple years old, had never made a profit, had no obvious chance of making a profit anytime soon, and had a higher market cap than General Motors. In other words, small Internet startup companies were thought to be more valuable than the world's biggest industrial company.
Companies are often grouped by their total price, their market cap. Thus we have large cap stocks, such as GM, IBM, GE, TI. We have mid cap stocks, which are middle sized companies with several hundred to a few thousand employees. And we have small cap stocks, which are very small companies, with perhaps 50 to 300 employees. It's often the case that the smaller cap stocks have more fluctuations in their price, resulting in more opportunities for profit and more opportunities for loss. Some people will prefer the risk and potential payback of small cap stocks, others will prefer the relative stability and lower risk of the large cap, or "blue chip" stocks.