If you run a business by yourself, then you are considered to be a sole proprietor or sole trader.
What this means
You alone are responsible for any debts your business may incur. You are also fully liable for any contracts or agreements your business has made. Legally, there is no distinction between you (as an individual) and your business. You receive all the profits (after taxation).
If a business is run by two or more people, and they each give a capital contribution (either in the form of property, money, or skill), it is called a partnership.
What this means
You and your partners have a joint responsibility for the business. Each one has his or her own share of the profit, depending on the agreed percentage of division. However, each individual will also be liable for the other's debts.
A corporation is owned by shareholders. Each individual is represented by his or her common stock. A board of directors is usually created to oversee management.
What this means
If you buy stock in a company, you become a part owner of the business. You have a share in the profits while having limited liability regarding debt, as the corporation is considered to be a separate legal entity from the individuals that run it.