An important distinction to understand upfront is the difference between a non-profit entity under state law as distinct from a tax-exempt entity under federal law.
Most non-profits are formed as non-profit organizations under state laws that deal specifically with such organizations. There are not shareholders or owners -- instead there are directors or trustees who control the entity for the benefit of the public or a defined class of members. So, there are no profits to distribute to any owners. Unless the non-profit also obtains tax-exempt status under federal law, though, it will still likely have to pay corporate tax on its net income.
This does not mean that a creator or founder cannot be paid a reasonable salary or fee for managing the non-profit. But this would be compensation for services rendered and not "profit" in the true sense of the word. In order to apply for and obtain tax-exempt status, it would also have to be shown that such payments would not violate prohibitations against private inurement.
The best known tax exempt status is probably as a public charity under Internal Revenue Code section 501(c)(3). Again, the basic steps are that you first have to form and duly organize a non-profit entity under state law. You then have to file a Form 1023 (Application for Recognition of Exemption under 501(c)(3) of the Internal Revenue Code) to seek tax-exempt status under federal law, and the IRS has to approve it.
You can pull up a copy of a Form 1023 on the IRS's website. Currently, there is a minimum $300 filing fee. You will see how complex the application process can be. It should help you, though, to develop a better understanding of what the requirements are for becoming a tax-exempt entity.
-- Thomas Pedreira