This will sound like one of those intelligence tests you hated in high school: How come you can buy a stock from the stock exchange, but you can't buy a stock ON the stock exchange? Don't worry; it's not as complicated as it sounds. The truth is you can buy any stock you want. Once you know where to go to buy it, the rest is done behind the scenes and doesn't really affect you.
Only a broker can purchase stocks on the stock exchange. That means you have to go to a broker and ask him to make the purchase for you. Back in the day, that meant actually going to a broker's office. Today, there are many online brokerage companies, which make it easy to make trades on your computer. Once you select your stock and click "buy," a behind-the-scenes broker takes it from there.
Once tell your broker what stock you want, how much of it you want, and what price you want to pay, your broker contacts a trader at the stock exchange. That trader lets the other traders know he has a "buy" order. Another trader agrees to sell your trader the stock. Then your trader contacts your broker to let him know the deal went through.
When you decide you want to sell the stock you bought, you have to undergo the whole process all over again. Get hold of your broker or press "sell" on the website you're on. The broker tells the floor trader to sell the stock. That trader finds a trader who wants to buy.
You might wonder how all of these people know what price you are willing to pay. You can put out an order for "market" price, which means you will take any price. If that makes you nervous, put out an order at a "limit" price. This tells people the most you are willing to pay to buy and the least you will take if you sell. The broker and the trader have to obey your wishes. This protects you from big losses.
You have to put money in an investment account with the broker you choose. You fill out basic information about yourself and get assigned an account number. You can then send money by mail or by electronic bank transfer to that account. Your agreement with your broker allows him to retrieve funds from that account when you want to buy.