IRS Tax Code Section 501(c)(3) is just one of the tax law provisions granting exemption from the federal income tax to non-profit organizations. This exemption does not cover other federal taxes such as employment taxes.
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.
Another provision, 26 U.S.C. § 170, provides a deduction, for federal income tax purposes, for some donors who make charitable contributions to most types of 501(c)(3) organizations, among others. Regulations specify which such deductions must be verifiable in order to be allowed (e.g., receipts for donations over $250).