All 501c3 organizations have what is called a "foundation classification." The terms "public charity" and "private foundation" are ways of referring to an organization's foundation classification. Because of the way the law is written, any organization that qualifies for tax-exempt status under section 501c3 is presumed to be a private foundation, unless it can show that it qualifies for one of the exceptions to private foundation status. Any organization qualifying for such an exception is sometimes called a public charity.
Some types of organizations, such as churches and schools, are defined as public charities by law. But most organizations qualifying for public charity status do so because they can show that their financial support comes from a broad cross-section of the public. Organizations that receive their support from a very narrow base, or that were set up by a wealthy individual or family will typically be classified as private foundations.
Although both types of organizations are tax-exempt under section 501c3, private foundations are subject to certain excise taxes and reporting requirements that do not apply to public charities.