A company can choose to offer a percentage of itself to the general public. This means everyone can invest in this company if they want to. When you want to buy a portion of a company you need to buy a share. A share is the name given to that portion of the company. This doesn't mean you own the company, it just means you have invested in it.
In Neopets, you can purchase stocks in the Stock Market game. Buy low, sell high!