A partnership is a business with two or more owners who conduct business and are not incorporated.
A partnership can exist for both tax and legal purposes without a formal, written agreement.
A partnership itself does not pay income taxes, but it does file an information return called Form 1065. The end results of Form 1065 flow through to a form called Form K-1, which is sent to each partner. The partners then report their share of partnership operations on their personal tax returns.
John and Joe are 50/50 owners of a partnership. For the year, the partnership had $100,000 of gross income and $40,000 of expenses, for ending net income of $60,000. These numbers will be reported on Form 1065. John and Joe will each receive a Form K-1 from the partnership, showing their 50% share of the net income (so $30,000 in this case). John and Joe will file their personal tax returns showing $30,000 of income from the partnership.
I wrote extensively about partnerships and the choice of business entity in a series of posts in 2015 and early 2016: