A partnership is a business owned by two or more people that hasn't filed papers to become a corporation or a limited liability company (LLC). You don't have to complete any paperwork to create your partnership -- the arrangement begins as soon as you start a business with another person.
Although the law doesn't require it, many partners work out the details of how they will manage their business in a written partnership agreement. If you don't create a written agreement, the partnership laws of your state will govern your partnership.
If you need legal advice for your business, talk to a lawyer.