Definition
A federal regulation that establishes the pilot in command as the person directly responsible for, and the final authority as to, the operation of the aircraft. It also permits the pilot in command to deviate from any rule in Part 91 to the extent required to meet an in-flight emergency, and requires a written report to the FAA on request when such a deviation occurs.
Plain English
This is the rule that says the pilot in command is in charge of the aircraft and is responsible for what happens during the flight. It also says that in an emergency, the pilot can break other Part 91 rules if needed to handle the situation, but may have to file a written report afterward if the FAA asks for one.
Context Anchor
You will see this citation in FAA handbooks, checkride discussions, and weather avoidance situations where ATC may provide assistance but the pilot still remains responsible for the aircraft.
Derivation
CFR stands for Code of Federal Regulations -- the organized collection of U.S. federal rules. 'Part 91' is the section covering general operating and flight rules, and 'section 91.3' is one specific rule within that part. So '14 CFR 91.3' simply points to one numbered rule inside the federal aviation regulations.
Why Pilots Care
It gives pilots the legal responsibility and authority to make safety-first decisions, including weather deviations, without fear of regulatory violation when acting in good faith.
Intuition Check
Do not read 14 CFR part 91 section 91.3 as just a page reference or a weather procedure. It is a legal rule that says who has final responsibility for the aircraft: the pilot in command.
Example Sentence 1
Citing 14 CFR 91.3, the pilot in command deviated from her assigned heading to avoid a rapidly building thunderstorm cell.
Example Sentence 2
Under 14 CFR part 91 section 91.3 the captain accepted full responsibility for the go-around decision when weather at the destination deteriorated.