Plain English Meaning
A situation dangerous enough that continuing to follow the normal rules could make things worse. When a flight is genuinely at risk, the pilot is allowed to break a rule if breaking it is what keeps the flight safe.
Full Definition
A regulatory phrase used in 14 CFR 91.3(b) describing conditions that pose a genuine threat to the safe conduct of a flight, serious enough to justify the pilot in command deviating from any rule of Part 91 to the extent required to meet that emergency.
Context Anchor
Seen in aeronautical decision-making when a pilot is deciding whether a situation is still safe enough to continue.
Derivation
Hazard comes from an old word connected with chance or risk. In , hazardous points to something more serious than ordinary difficulty: it is a risk that can threaten the of the flight.