Corporation, and
Non-Profit Organization. Lets start with Sole proprietorship.
Sole Proprietorship is when you start your own business with permission from the government. You use the governments prices and get to sell your own products.

Partnership is when you join with another person to help you run a business. If you wanted to make a business and you didn't have enough money you get a partner. See its that easy.
Corporation is when you have stocks in the business. Like what we do in our stocks project thats part of a corporation business.
Non-Profit Organization is when you don't always use your products to get money for your business. It dose not distribute its products to share holders.