A non-profit organization (NPO) is a legally constituted organization whose primary objective is to support or to actively engage in activities of public or private interest without any commercial or monetary profit purposes. NPOs are active in a wide range of areas, including the environment, humanitarian aid, animal protection, education, the arts, social issues, charities, early childhood education, health care, politics, religion, research, sports or other endeavors.
They may be organized as trusts, foundations, unincorporated associations or other structures.

There are several considerations which may or may-not apply depending on the type of NPO:
Exemption from taxes (corporate, property and excise)(federal, state)
Charitable tax deductions for donors
Legal requirements: Financial reporting, distribution of assets, limits on political activities.

Tax-Exempt Status:
Non-profit organizations are exempt from some Federal income taxes.

They may have to pay tax on income derived from activities unrelated to their exempt purpose.
See: Tax-Exempt Status

Tax-deductibility:
Contributions to 501(c)(3) organizations (Charities - see below), except organizations testing for public safety, are deductible as charitable contributions on the donor's federal income tax return.
Donations to most other NPOs are not deductible
Fraternal Lodge Societies [501(c)(10)] donations may be deductible if used for 501(c)(3) purposes. e.g. The Shriners fund for the Shriners Hospital for Children.

Contributions to Churches are automatically deductible even if they weren't a 501(c)(3).
S See: Facts about 501c3

Charities [IRS 501(2)(3)] are a type of NPO which are tax-exempt and allow charitable deductions for donors.
501(c)(3) exemptions apply to corporations, and any community chest, fund, or foundation, organized and operated exclusively for religious, charitable, scientific, testing for public safety, literary, or educational purposes, or to foster national or international amateur sports competition, or for the prevention of cruelty to children or animals.
Individuals giving to 501(c)(3) organizations that are private foundations may generally deduct contributions representing up to 30% of their adjusted gross income. Private foundations may deduct contributions representing up to 50% of the donor's adjusted gross income.

Organizations with this classification are prohibited from conducting political campaign activities to influence elections to public office. Public charities (but not private foundations) are permitted to conduct a limited amount of lobbying to influence legislation.

See: Exemption Requirements article at the IRS.

Trusts:
Charitable Remainder Trusts (CRT) and Charitable Lead Trusts (CLT) are commonly used for estate planning purposes.

Other Types of Non-profits Tax Information for Charities & Other Non-Profits (www.irs.gov/charities/) at the Internal Revenue Service (IRS) describes several types:

* Charitable organizations (IRS Code section 501(c)(3))

• Public charities - Usually broad support with income from the public or government.
    Examples are: American Red Cross, Habitat for Humanity, United Way, Salvation Army, University funds, Churches
  
• Private foundations - Receives most of its income from investments and endowments. e.g. Family such as the Ford Foundation ($12 B), the Robert Wood Johnson foundation ($9 B), the William and Flora Hewlett Foundation ($8 B), the Bill and Melinda Gates foundation ($28 B)
  
• Most churches in America have organized as "501c3 tax-exempt religious organizations." This is a fairly recent trend that has only been going on for about fifty years. Churches were only added to section 501c3 of the tax code in 1954.
• Between 1996 and 2003, eleven pharmaceutical manufacturers established operating foundations for the purpose of distributing medications to patients with financial hardships. E.g. GlaxoSmithKline Patient Access Programs Foundation, Lilly Endowment, ...
• A donor advised fund program allows individuals, private foundations, and business entities to make irrevocable contributions to the public charity sponsoring the program which are then allocated to a specific account, the individual "donor advised fund," from which donor advisors or account holders may then recommend grants to qualified grant recipients on their own timetable.
  See more below.

* Labor organizations and farmer co-ops (section 501(c)(5))
* Business leagues (Chamber of Commerce orgs, trade associations), professional football leagues (e.g., the NFL) (section 501(c)(6))

501(c) Organizations at Wikipedia describes 27 types.

More on Donor advised funds:
When donors make contributions, they may in turn be eligible to take tax deductions. They also receive and obtain certain privileges, including that of recommending grants from the donor advised fund to IRS-qualified 501(c)(3) public charities and certain private operating foundations. The charity will generally perform due diligence to verify that each nonprofit organization to which a grant is recommended is an IRS-qualified 501(c)(3) public charity or qualified private operating foundation, and will further verify that that the grant is compliant with the policies of the charity and its donor advised fund program. The Fidelity® Charitable Gift Fund is one of the nation’s largest ($2.2 B) public charities and has the largest donor-advised fund program in the country.

Donor advised funds have an advantage over setting up your own 501(c)(3) in that you don't have to do the paperwork to establish it, file yearly returns (Form 990 Return of Organization Exempt From Income Tax), establish a board of directors and officers (President, Secretary and Treasurer), or make minimum distributions.

The IRS is looking into making these more restrictive, e.g. requiring a minimum (e.g. 5%) distribution.

See: Start your own charitable foundation - USA Today Article
Charitable Gift Trusts at fool.com

IRS provisions:
501(a) - Tax-exempt status.
509(a) - A private foundation as defined.

Links:
Charitable Donations - Philanthropy - fundraising
Evaluating non-profits at the Society area
Tax Information for Charities & Other Non-Profits (www.irs.gov/charities/)
501(c) Organizations at Wikipedia
Exemption from tax at Cornell Law
Fund Raiser / Grant-Writing Consultants
A Primer on Online Fundraising for Nonprofit Organizations at TechSoup.org
Tips on Tax Deductions at give.org by the BBB
Charitable Donation Deductions: IRS Tax Deductions for Charitable Contributions at NonProfitManagement.suite101.com

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