Plain English Meaning
The highest altitude where the airplane can still climb at a useful, defined rate. Above this altitude the airplane can still go higher, but only very slowly.
Full Definition
The maximum density altitude at which an airplane can still climb at a specified minimum rate under standard atmospheric conditions. For a piston-engine airplane, that minimum rate is 100 feet per minute; for a turbine-engine airplane, it is 500 feet per minute.
Context Anchor
Seen in climb performance discussions and performance charts, especially when comparing service ceiling with absolute ceiling.
Derivation
Service' here means 'practical use.' The service ceiling is the altitude up to which the airplane is still useful for climbing — above it, the climb rate becomes too small to be operationally meaningful.