In order to act as Pilot in Command of an aircraft a pilot must have three things in their possession: Medical, Photo ID, and Pilot’s Certificate. Per §61.51 student pilots must carry their logbook on solo cross-country flights and sport pilots must carry logbooks (or other evidence of endorsements) on all flights.
§61.3 Requirement for certificates, ratings, and authorizations has the details, but the important parts for most pilots are below:
(a) Required pilot certificate for operating a civil aircraft of the United States… in the person’s physical possession or readily accessible in the aircraft when exercising the privileges of that pilot certificate or authorization
(2) Has a photo identification that is in that person’s physical possession or readily accessible in the aircraft when exercising the privileges of that pilot certificate or authorization. The photo identification must be a:
Acceptable photo ID includes a driver’s license, government identification card, passport, or credential that authorizes unescorted access to a security identification display area at an airport regulated under 49 CFR part 1542.
(c) Medical certificate. (1) A person may serve as a required pilot flight crewmember of an aircraft only if that person holds the appropriate medical certificate issued under part 67 of this chapter, or other documentation acceptable to the FAA, that is in that person’s physical possession or readily accessible in the aircraft.
Beginning May 1, 2017 (assuming the current administration doesn’t bollix up the works) pilots may fly with just a driver’s license, if you comply with the medical rules and are flying under Part 91 in a plane with 6 or fewer seats.
BasicMed – 3rd Class Medical Reform