There are many different forms of corporations. The traditional corporation is what is traded on the open stock market- anyone can buy shares and their percentage of shares gives the share holder that much voting power. The day to day decisions are made by a board of trustees and the corporation may or may not pay dividends to it's shareholders. The next variation is the closely held corporation, same idea but the shares are sold privately.
For a small business owner, there are three options, LLC, S-corp and C-corp. The obvious advantage is, if you incorporate the corporation becomes its own entity and if for some reason you default on the loan or get sued they can't go after your personal assets- they can only go after the CORPORATION'S assets. You only stand to lose what you personally invested out of pocket. It acts like a shield protecting your assets (your personal house, car, money in the bank, etc.)
S-corp's and C-corp's are a little more complicated in terms of documentation, taxes, etc. C-corp's are subject to double taxation- any profits the corporation makes gets taxed and if you decide to pull money out of the corporation (i.e. pay yourself a dividend) it gets taxed again. An S-corp requires you to pay yourself a 'reasonable salary' so if you're trying to keep yourself under a certain income bracket or something like that this may or may not work for you. An LLC (limited liability corporation) doesn't really change the taxes- you will be taxed the same as a sole proprietorship (just you owning the business) or partnership (if you have a business partner) BUT it still protects your personal assets. Its also simpler in terms of taxes and documentation.
I suggest talking to an accountant and/or tax specialist to see which is the best option for you.
**Here's a fun fact to leave you with- you do not have to incorporate your business in the state in which you live or the state where your business physically exists. Shop around and see which state offers the BEST deal. For example Delaware offers some really good rates. You can incorporate your business in Delaware without ever setting foot there.