The term for the time measured by a clock moving along its own worldline is proper time. It is the time recorded by a clock that travels between two events, such as a wristwatch carried by an astronaut or a clock attached to a spacecraft.
Proper time is a central idea in special and general relativity. It is determined by the path through spacetime, not merely by the two events themselves. Two clocks can begin together and reunite later having recorded different amounts of time if they followed different paths or experienced different motion and gravity.
A common mix-up is confusing proper time with coordinate time. Coordinate time depends on the chosen reference frame, while a clock measures its own proper time locally. The difference is behind the famous twin paradox and the corrections required for satellite navigation systems.
Proper time is defined for timelike paths, the kinds of paths that material observers and massive particles can follow. Light travels along null paths, for which the proper-time interval is zero.