501c3 (without tax exemption)

The basic requirement of obtaining tax-exempt status is that the
organization is specifically limited in powers to purposes that the IRS
classifies as tax-exempt purposes. Unlike for-profit corporations that
benefit from broad and general purposes, non-profit organizations need
to be limited in powers to function with tax-exempt status, but a
non-profit corporation is by default not limited in powers until it
specifically limits itself in the articles of incorporation or nonprofit
corporate bylaws. This limiting of the powers is crucial to obtaining
tax exempt status with the IRS and then on the state level.  Organizations acquire 501(c)(3) tax exemption by filing IRS Form 1023.  As of 2006
the form must be accompanied by a $850 filing fee if the yearly gross
receipts for the organization are expected to average $10,000 or more.  If yearly gross receipts are expected to average less than $10,000, the filing fee is reduced to $400.
There are some classes of organizations that automatically are treated
as tax exempt under 501(c)(3), without the need to file Form 1023:

• Churches, their integrated auxiliaries, and conventions or
  associations of churches. A convention or association of churches
  generally refers to the organizational structure of congregational
  churches.
  A convention or association of churches can also refer to a cooperative
  undertaking of churches of various denominations that works together to
  perform religious activities.
• Organizations that are not private foundations and that have gross receipts that normally are not more than $5,000

The IRS released a software tool called Cyber Assistant in 2013, which was succeeded by Form 1023-EZ in 2014.

There is an alternative way for an organization to obtain status
if an organization has applied for a determination and either there is
an actual controversy regarding a determination or the Internal Revenue
Service has failed to make a determination. In these cases, the United States Tax Court, the United States District Court for the District of Columbia, and the United States Court of Federal Claims have concurrent jurisdiction
to issue a declaratory judgment of the organization’s qualification if
the organization has exhausted administrative remedies with the Internal
Revenue Service.

Prior to October 9, 1969, nonprofit organizations could declare
themselves to be tax-exempt under Section 501(c)(3) without first
obtaining Internal Revenue Service recognition by filing Form 1023 and
receiving a determination letter.
A nonprofit organization that did so prior to that date could still be
subject to challenge of its status by the Internal Revenue Service.

The two exempt classifications of 501(c)(3) organizations are as follows:

• A public charity, identified by the Internal Revenue Service
  (IRS) as “not a private foundation”, normally receives a substantial
  part of its income, directly or indirectly, from the general public or
  from the government. The public support must be fairly broad, not
  limited to a few individuals or families. Public charities are defined
  in the Internal Revenue Code under sections 509(a)(0) through 509(a)(4).
• A private foundation,
  sometimes called a non-operating foundation, receives most of its
  income from investments and endowments. This income is used to make
  grants to other organizations, rather than being disbursed directly for
  charitable activities. Private foundations are defined in the Internal
  Revenue Code under section 509(a) as 501(c)(3) organizations, which do
  not qualify as public charities.