Plain English Meaning
The strongest setting the engine is allowed to run at for as long as you want without damaging it. Takeoff power is stronger but can only be used briefly; this is the highest setting that's safe to leave on continuously.
Full Definition
The highest power setting an engine is approved to sustain indefinitely without time limit, as established by the manufacturer and reflected in the aircraft's operating limitations. It is lower than takeoff power, which is typically time-limited to a few minutes.
Context Anchor
Seen in aircraft operating limitations, engine specifications, and performance discussions for light-sport aircraft.
Derivation
Continuous' comes from Latin continuus, meaning 'uninterrupted' or 'unbroken.' The term literally describes the maximum power the engine can produce on an unbroken, time-unlimited basis -- distinguishing it from short-duration like takeoff power.