What are Stocks?
Corporate stocks are units of ownership in that company. You have 'equity' in the company (stocks are also called 'equity securities'), which means you also have the right to share in the company's earnings through dividend payments, and the company's growth from the appreciation of the stock prices.
How much equity of the company you own is determined by how many shares you own, compared to how many shares of the company stocks are outstanding. You can find this out by looking at the company's information (or perspectus), which you can look up at various links (or go directly to the SEC itself here:SEC EDGAR Archives. Just type in the company's full name at the search engine to get all the company's SEC filings).

Companies issue stocks and bonds to raise money from public investors. This money is usually needed for specific company goals, such as new facilities, new research and development, increasing sales by expanding operations, or even repaying debts. Issuing stocks means that the company is giving up part of the ownership of the company in return for the investment capital.
There are 3 types of capitalization for a public corporation:
Common Stock (which is what we refer to on this site when we say 'stocks') is the most widely traded of company stock. It is usually liquid (the exception is common stock in a company which is very infrequently traded on the exchanges; these stocks may require some time to find either a seller or buyer for).
Preferred Stock, which is almost a cross between common stock and a company issued bond because it usually offers a fixed rate of interest, yet also represents ownership in the corporation, though there are no voting rights. There are several different types of preferred stock, but since this is a discussion of basic stock market trading, we aren't going to go into details about these.They are also limited in many ways.
Corporate Bonds. These do not represent ownership in the company. They are usually issued for a fixed time period with an expiration date. Essentially these represent loans to the company with fixed payouts and fixed maturity dates, after which the debt is no more.

When a corporation 'goes public', which means issues shares available for trade to public investors, the company must register with the Securities and Exchange Commission (SEC), so their company information becomes public knowledge and publicly available. It is this information that we need to look at to help determine whether a stock is something we want to invest in, in addition to the news and rumors available all over. Because this company information is very strictly monitored for accuracy by the SEC, and corroborated by an independent accounting firm, it is the most accurate guide to a company's well being.

When you buy 'stocks', we are talking about common stocks traded on the public exchanges, such as the NYSE or AMEX or NASDAQ (the list of stock exchanges are US Stock Exchanges).
Common stockholders have certain rights:
1. Receive Dividends. If a company declares dividends, then each share of stock is entitled to the same amount of dividends as any other share. A company does not have to declare dividends however, even if it has profits and earnings in excess of expenses. Dividend payment is entirely dependent on the Board of Directors of that company. The Board of Directors may choose to reinvest their profits, which is often the case with 'growth' stocks.

2. Vote. You have the right to vote for Board candidates or company policy based on the number of shares you own. There are various different methods for this, the best way to find out the details for any given company is to read the company prospectus.

3. Proportionate Interest in the Company. This gives you the right to maintain your percentage interest in the company. If say the stock splits, your stock will be split as well. If you own 100 shares of company A's stock, after a 2 for 1 split, you will own 200 shares.

4. Limited Liability and Last Claim. You are not liable for the company's losses beyond what you paid for the shares of stock. If a company goes bankrupt, no one can come after any of your personal assets. Conversely, you do have a right to claim your loss against the company's assets should they go bankrupt. But you, as a common stock holder, are the last in line of claims. This means you will be the last paid if a company is liquidated. In reality, if this happens, you most likely will have lost all that you've paid for the stock.

How does the Stock Market work?
When you place your order for a listed stock with a broker, the broker records the order on an order ticket. This ticket is electronically transmitted to the floor of the Exchange to the brokerage's representative there. At the Exchange, the 'floor broker' (an Exchange member who executes orders for your brokerage) takes the order to the 'trading post' (the physical location where that particular stock trades), and executes it with another floor broker or the stock 'specialist' (an exchange member firm located at that trading post who ensures that each stock transaction is covered). The order is then completed. The price is recorded and reported on the ticker (the electronic display on the floor, or what you see on CNN during trading hours, which continally shows the stock symbols and new prices as they are executed). Confirmation of your order is then sent to your brokerage, who is responsible for notifying you. You will be notified of the price you bought/sold at and the number of shares. Your transaction is now complete.
Depending on the activity of the stock on the trading floor, you may not buy or sell at the price you were quoted. It is also apparent that there could be delays filling your order due to the process involved. Therefore, you do not consider a trade done til the broker has sent you confirmation. When a stock is trading fast and furiously on the floor, you could buy in at a substantially higher/lower price than you expected.
There are 2 different order time frames. The day order(the order is only good until the end of the trading session of the day it was placed. If not filled or cancelled, it expires.) or the GTC (good til canceled). ALL orders are assumed to be day orders if you do not specify GTC. If a GTC order isn't filled in a day, it is a standing order until it can be filled. If you place this type of order, you should talk to your broker about their GTC policies and how long they will keep them on their systems for.

There are several different kinds of orders. Most investors use only a few. Here are the most frequently used types:
Market Order - this is the most common one. If you don't put restrictions on your order, all orders are executed as market orders. You specify the number of shares and the stock, but not the price. The ordering format for a market order is: "buy (number of shares) XYZ stock". The stock order is filled at the current price of that stock when the order hits the trading floor. Market orders are filled promptly if it's a commonly traded stock.
It is understood that a market order is filled at the lowest available price for a buy order, and the highest available price if it's a sell order. No price is guaranteed on a market order, it is determined on the floor at the time of a trade. For example, when AOL moved 40 points in a day, some people who got quotes from their brokers got their orders filled at prices substantially higher than the quote they saw. This is because the price was moving up continuously, and the broker could not get the order in ahead of others also placed. There is always a time delay from when you see or hear the quote to the time the order hits the floor for execution. That is the chance you take with a market order. However you know that if it's a stock commonly traded, your order will be filled. This is not the case with other order types.

Limit Order- you place this when you specify the price you want your order to be filled at, in addition to the stock and number of shares. You would specify the order as "buy (number shares) of XYZ stock at (price)".
Your order must be filled at or better than the price you specify. The limit price is never the same as the stock's current quoted price. You give this price because you expect the stock to move in the direction of your specified price, and at that point, you want your order executed. If you don't specify GTC with this type of order, it is cancelled at the end of the day's trading session.
You can place a buy limit order or a sell limit order. A buy limit order is placed to buy a stock below it's current price. This order may never get filled if the stock goes up. The reason why you don't bother giving a limit on an order higher than it's current value is because it would get filled at market immediately anyway. Sell limit orders are the same, except the price is higher than it's current price. If the stock goes up to your limit, the transaction is automatically executed.
Never forget that you have placed a limit order GTC. I once placed a GTC sell limit order on a stock and forgot about it. The stock rose phenomenally due to the release of their new product. My standing sell order executed the stock sale at $9, while the stock price continued that same day up to $28. What a lost opportunity!

Stop Order- or stop loss order. This is used to limit losses on stock positions in case prices fall suddenly or sell when a price hits an acceptable profit limit. Stop orders are NOT accepted on NASDAQ or OTC markets. When a stock price hits the stop order price, the order becomes a market order and it WILL be executed. This is different from a limit order, which may not be executed because the window of pricing opportunity has passed. However, with a stop order, the order may not be executed at the stop price because of it's change to a market order. You don't know what price your order was filled at until you receive confirmation from your broker. What this type of order is good for however is if you don't have the time to keep close watch on your portfolio and have very specific loss/profit goals. When your stock hits the goal, you accept that selling or buying the stock is a good idea, regardless of other market conditions.