According to Wikipedia, a 501(c) organization, also known colloquially as a 501(c), is a tax-exempt nonprofit organization in the United States. Section 501(c) of the United States Internal Revenue Code (26 U.S.C. § 501(c)) provides that 29 types of nonprofit organizations are exempt from some federal income taxes.
Following is an excerpt posted by Cornell Law School that pulls out the salient parts of the 501(c) article itself.
Following is an article from the Internal Revenue Service that covers the tax and record keeping ramifications for a 501c non-profit.