A nonprofit organization is an organization whose goal do not include a personal profit for its owners or organization. Nonprofit organizations often do strive for internal gains, but they use them to meet their social or educational goals rather than for personal profit. 
Ownership is the quantitative difference between for- and not-for-profit organizations. For-profit organizations can be privately owned and may re-distribute taxable wealth to employees and shareholders. By contrast, not-for-profit organizations do not have private owners. They have controlling members or boards, but these people cannot sell their shares to others or personally benefit in any taxable way. 
A nonprofit organization is formed by incorporating in the state in which they expect to do business. The act of incorporating creates a legal entity enabling the organization to be treated as a corporation under law and to enter into business dealings, form contracts, and own property as any other individual or for-profit corporation may do. To be exempt from federal income taxes, a nonprofit organization must meet the requirements set forth by the Internal Revenue Service (IRS).
IRS Publication 557 contains the types of nonprofits that are available.
AcqLinks and References:
-  Nickels and McHugh (2010) “Understanding Business” McGraw-Hill Irwin
-  Website: Wikipedia – Nonprofit organization
- IRS Publication 557 – Tax-Exempt Status for Your Organization
- California Nonprofit Publication
- Website: Internal Revenue Service (IRS)
- Website: Answer.com – The Guide to Taxes