The answer depends on state law.
For example, Illinois is governed by 625 ILCS 5/6. Generally, people need an instruction permit to participate in a driver education course.
Section 6-103. The Secretary of State shall not issue, renew, or allow the retention of any driver's license nor issue any permit under this Code:
To any person, as a driver, who is under the age of 18 years except as provided in Section 6-107, and except that an instruction permit may be issued under Section 6-107.1 to a child who is not less than 15 years of age if the child is enrolled in an approved driver education course as defined in Section 1-103 of this Code and requires an instruction permit to participate therein, except that an instruction permit may be issued under the provisions of Section 6-107.1 to a child who is 17 years and 3 months of age without the child having enrolled in an approved driver education course and except that an instruction permit may be issued to a child who is at least 15 years and 3 months of age, is enrolled in school, meets the educational requirements of the Driver Education Act, and has passed examinations the Secretary of State in his or her discretion may prescribe;
1.5. To any person at least 18 years of age but less than 21 years of age unless the person has, in addition to any other requirements of this Code, successfully completed an adult driver education course as provided in Section 6-107.5 of this Code
It is written in this way because 1) IL has what's called a graduated license program where minors who would otherwise be allowed to get a license may only get one subject to parental approval, and 2) sometimes people apply for licenses after moving to IL.
What do you need a permit or license for?
Section 6-101. Drivers must have licenses or permits.
(a) No person, except those expressly exempted by Section 6-102, shall drive any motor vehicle upon a highway in this State unless such person has a valid license or permit, or a restricted driving permit, issued under the provisions of this Act.
(b) No person shall drive a motor vehicle unless he holds a valid license or permit...
There are a few exemptions, one of which is for temporarily driving farm equipment along a roadway.