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A student pilot certificate is issued to a pilot in training, and is a pre-requisite for the student to fly alone in the aircraft, or solo. It is usually initially issued by a medical doctor who is also an authorized aviation medical examiner, in conjunction with the student's first medical certificate.
A private pilot is the holder of a private pilot license. With an FAA-issued private pilot license (PPL), a person is able to fly US registered aircraft that meet the same aircraft category and class that their private pilot license designates.
A commercial pilot license (CPL), is a qualification that permits the holder to act as a pilot of an aircraft and be paid for his/her work.