For Profit Schools
A for profit school is controlled by the owner. It is usually in business to make money or turn a profit. Parents pay for the school's services just as though they were customers. Examples of this sort of school include Le Rosey in Switzerland, Sylvan Learning Centers, the Nobel Schools, as well as thousands of small nursery and primary schools. The school can be incorporated or a sole proprietorship or some other form of ownership.
The advantage is that the owner truly calls the shots and controls everything. If the parents do not like what is being offered, they can take their children elsewhere. Parental input may be sought and even entertained, but the final decisions and control rest with the owner.
Not For Profit Schools
Most private schools seek 501(c)(3) status from the IRS once they have incorporated. This is the non-profit status which exempts the entity from federal, state and local taxes. It also permits the school to accept contributions which are in turn tax-deductible to the extent provided by law.
Some schools are set up so that parents own shares in the corporation. The ownership formulae vary but most seem to depend on the number of children enrolled in the school.
Most not for profit schools will have an active board of trustees. The trustees are the mechanism by which the school governed. Depending on how the charter is set up, its powers will be quite extensive. A headmaster at such a school is hired by the board of trustees. This governance mechanism allows the school to outlive its founder, and, all things being equal, last for generations. Examples of this sort of school include Kent School which was founded by Father Frederick Herbert Sill in 1906 and Exeter Academy which was founded by John Phillips in 1781.