A sole proprietorship is a form of business that is legally indistinguishable from the person running it. Also known as a DBA (doing business as), a sole proprietorship is the cheapest and easiest type of business to create. Just get a business license, hang your shingle, and you're a sole proprietor.
A partnership is a business run by two or more people. Although you don't have to have a written partnership agreement to start a partnership, it is best to have one.
Unlike a sole proprietorship and a partnership, a corporation is a legal entity separate from the principals in the business. The great thing about that is the corporation protects you from personal liability for business debts. For example, if the business gets sued, the plaintiff can only go after business assets — not your personal assets.