I am a member of the board of a 501(c)(3) horseback riding facility. Another facility near us calls itself a "nonprofit" organization on its web site but is not on the government's list of 501(c)(3) organizations. Can an organization advertise itself as a nonprofit, but not be a 501(c)(3) organization?
Yes. Nonprofit is a concept of state law, which means that surplus revenue cannot be distributed to owners or other insiders. When we say “nonprofit” we usually think of federally tax-exempt organizations, but Section 501(c) of the Tax Code, which describes most, but not all, exempt organizations, now lists 29 separate types of organizations that are tax exempt.
The largest class by number is 501(c)(3) charities, but the statute also includes (c)(1) groups like federal credit unions, (c)(4) civic associations, (c)(5) unions or agricultural associations, (c)(6) trade associations like the National Football League, (c)(7) social clubs, various fraternal organizations like the Masons or the Moose, veterans organizations and a lot of others. Most of these are nonprofit organizations. There are also nonprofits that are not exempt, such as, until a few years ago, the New York Stock Exchange. (See Ready Reference Page: “What Do We Mean When We Say Nonprofit?”)