The term, nonprofit, is used rather loosely to describe groups that come together to achieve a mission that involves serving the public good, rather than to make a profit.
The term "nonprofit" does not imply any specific type of legal structure. If the group incorporates, it is a nonprofit corporation. If it does not incorporate, it is an unincorporated nonprofit association.
Examples of unincorporated nonprofit associations could include a group of neighbors who want to keep a neighborhood park clean and kept up, or a group of soccer dads who sell candy to fund their children's trips to soccer tournaments.
Both groups could be called "nonprofits" simply because they are mission-driven, not profit-driven. Let's say the park group never does anything to create a formal structure. It would be considered an unincorporated nonprofit association.
The soccer dads take another route and file papers with the state to become a corporation. The group is then a nonprofit corporation. As a result, this group, because it works for the "common good," would be exempt from the state taxes a for-profit corporation would have to pay, making it, to some extent, publicly subsidized.
To make sure that the nonprofit corporation actually works for the public good and earns its tax breaks, the state will required it to set up certain organizational structures such as a board of directors or board of trustees. The board is responsible for keeping the group on track and fulfilling its nonprofit mission.
Any nonprofit, the unincorporated and the incorporated, could become tax-exempt from federal income taxes by applying to the IRS. However, incorporated nonprofits stand the best chance of actually meeting the IRS standards for tax-exemption.
There are several types of federal tax-exempt status, but the most favorable one is 501(c)(3) status. A 501(c)(3) nonprofit is exempt from federal taxes but, in addition, donors to the nonprofit can take a tax deduction for the amount of their donations. This is a powerful incentive for donors.
Other classifications offer exemptions from federal income taxes (for instance 501(c)(4) or 501(c)(6) status), but only 501(c)(3) status allows donors to deduct contributions.
As you can see, the very first decision any group will need to make is what kind of nonprofit it wants to be.