There are several certificates and documents that both the plane and the pilot need to have in order to be legal to fly.
PHAK Ch-01 (16) Pilot Certifications
61.23 Medical certificates: Requirement and duration.
61.3 Requirement for certificates, ratings, and authorizations.
91.203 Civil aircraft: Certifications
91.205 Instrument and equipment requirements
Private Pilot ACS Page 3 & 4 Pilot Qualifications & Airworthiness Requirements