How a system that can facilitate one billion shares trading in a single day works is a mystery to me, so I thought I'd do a little digging and see if I could come up with the general process on how stock trading works. Here is what I found.
The purpose of a stock market is to facilitate the exchange of securities between buyers and sellers, reducing the risks of investing. Just imagine how complicated it could become to sell shares if you had to search for someone who wanted to buy your stock.
When it comes to trading stocks, it's not trading in the typical sense of the word, where I give you this, if you give me that. Trading on the stock market is more a matter of buying and selling, these two components are what make up ‘trading' when it comes to investing in the stock market. As opposed to something like retail shopping where the prices are set by the seller and you can just walk into a store and purchase something, the stock market functions more like an auction in which both buyers and sellers are actively setting the prices at the same time.

With both buyers and sellers actively setting the prices in the stock market, it is only logical that there are subsequently two prices associated with every stock, the bid price and the ask price. The bid price is the price at which buyers are willing to buy the security whereas the ask price is the price at which sellers say they will sell the security. These two prices are pretty much never the same: generally, the bid is slightly below the ask. The difference between the two is called the spread, the amount that is taken by your broker as profit. Specialists, who are in charge of the coordination of the buying and selling of a certain stock, pair bids and asks together to streamline the process and keep the spread small, but positive.

Considering that the bid and ask prices are always changing, you need to be careful about your sales and purchases. The price that is quoted may or may not be the price at which you actually buy or sell the stock. There are several options regarding the method of execution for your trades:
Market Orders: an order to buy or sell stocks at the prevailing market price. These are often the lowest-commission trades because they involve very little work on the broker's part.
Limit Order: You tell your broker to buy a security at or below a specified price, or to sell a security at or above a specified price. This ensures that you will never pay more for the stock than whatever price you set as your "limit."
Stop Order: You tell your broker to buy a security at the market price once it reaches a level higher than the current market price. The opposite would be true if you were selling: you would tell your broker to sell your security once it reaches a level below the current market price. A market order to buy or sell a certain quantity of a certain security if a specified price (the stop price) is reached or passed.
Day Order: You tell your broker to execute the trade by the end of the day; otherwise, he or she does not fill the order.
All or None: an order type for a broker to execute a trade only if every share of an order can be filled in its entirety, or else not at all.
Fill or Kill: You tell your broker to execute the trade immediately; if the trade is not filled right away then your broker does not execute the order.