"Non-profit status is a state law concept. Non-profit status may make an organization eligible for certain benefits, such as state sales, property, and income tax exemptions. Although most federal tax-exempt organizations are non-profit organizations, organizing a non-profit organization at the state level does not automatically grant the organization exemption from federal income tax. To qualify as tax-exempt from federal income tax, an organization must meet requirements set forth in the Internal Revenue Code." – IRS website, Applying for Exemption FAQ #1
Note that tax-exempt, from the IRS perspective, means that an organization is exempt from paying federal corporate income tax on income generated from activities that are substantially related to the purposes for which the entity was organized (and for which the organization received tax-exempt status). The organization must still pay federal corporate income tax on income generated from activites which are not substantially related to its exempt purposes (i.e., unrelated business taxable income).
Further note that tax-exempt status on the federal level does not automatically make an organization tax-exempt on the state level. In California, an organization will need to receive a determination letter from the Franchise Tax Board in order to be exempt from state corporate income tax.