This is another example of the problems with the language of the nonprofit world.  The word “foundation” has no specific legal meaning and means many things to many people.  The word “nonprofit” has a specific meaning but has multiple other meanings in the public dialogue.  (See Ready Reference Page: “What Do We Mean When We Say ‘Nonprofit’?”)  

There is no dichotomy between corporation and foundation.  A corporation is the type of entity.  A foundation, as you are using the term, is a type of entity that is exempt from federal income tax. 

If you are already registered with your state, you probably registered as a nonprofit corporation.  If so, you are already a “nonprofit” organization.  You don’t have to apply separately for that status.  You are already precluded from paying dividends and sharing your profits with owners.  You have to apply separately for exemption from federal income tax, which is the expectation of many who use the term nonprofit interchangeably with tax-exempt.

If you are applying for tax-exempt status, one of the many types of tax-exempt entities is a charity.  You could also be recognized as a charity if you were another type of entity, such as a trust, an unincorporated association, or a limited liability company.  You can be recognized as a charity if your organization is organized and operated exclusively for charitable purposes and meet a few other requirements of section 501(c)(3) of the Tax Code. 

If you are recognized as a charity, in general, you will be further classified as either a public charity (sometimes called a public foundation) or a private foundation (which is an actual legally defined term), depending on whether you have broad public support in the form of gifts or purchase of goods or services.  You will normally be classified as a private foundation if you don’t have broad-based public support or a close relationship with a publicly supported organization.  There are both public charities (like the Arthritis Foundation, or a community foundation) and private foundations (like the Gates Foundation or the Ford Foundation) that use “foundation” in their name.  There are also some for-profit organizations that use the term in their name.  Some “foundations” are primarily grant makers supporting other charities; some run their own programs without giving any money away. 

You can be a corporation and a foundation, whatever the latter word means to you.