In corporations, owners are issued shares of stock and are called "shareholders" (or "stockholders" - they mean the same thing). A shareholder's shares are also sometimes referred to as their "equity" in the corporation.
For simplicity, and because it's the right choice for the vast majority of new businesses, Startomatic only allows you to issue common stock. There are many other types of stock (preferred, nonvoting, etc.), but common stock is the simplest and most, well... common. If you need to create different classes of stock in the future, you can do that with the help of a business lawyer.
A corporation's formation document (usually called Articles of Incorporation or Certificate of Incorporation) specifies the maximum number of shares of stock that may be issued, or its "authorized shares". The shares that are actually issued to the shareholders are called "issued shares".
In most cases, Startomatic defaults to 200,000 authorized shares; however, this tool allows you to change the number of authorized shares if you like. (Note that for Delaware corporations, we default to 5,000 authorized shares in order to save you money on filing fees and annual Delaware franchise taxes. You can change this if you like, but be aware of what a large number of authorized shares in Delaware may mean for your annual taxes and fees.)
It is also important to keep in mind that the shareholder or shareholders who own a majority of the shares of stock of of the company (i.e., more than 50% of all of the issued shares) usually control the election of the company's board of directors, and the directors control the major decisions of the company.