proper time
Proper time is the time that actually elapses on a clock carried along a particular path through spacetime, the time you would read off your own wristwatch on the journey. Unlike the time-readings that different observers assign to distant events, proper time is something everyone agrees on: it is the personal, lived time of the traveller. It is the invariant heart that survives beneath the shifting numbers of time dilation.
Different observers, slicing space and time differently, will assign different durations to the trip between two events. But they will all compute the same proper time for a given clock, because it is read directly from that one clock's own ticks. Proper time is therefore an invariant, a quantity that does not depend on who is doing the measuring, and it is built straight out of the spacetime interval between events.
Here lies a beautiful twist on intuition. Between two events that a single inertial clock can attend, the straight, unaccelerated path through spacetime racks up the MOST proper time. Every detour, every burst of acceleration, makes a clock age less, not more. This 'longest aging along the straight path' is the precise sense in which the stay-at-home twin ends up older than the traveller.
Imagine two friends driving between the same two towns, one taking the straight highway and one weaving all over the back roads. The wandering driver clocks more kilometres on the odometer. Spacetime works in a mirror-image way for time: the one who 'wanders' through space, by speeding up and turning around, logs LESS time on the clock, because the geometry of spacetime counts the time direction with an opposite sign.
Proper time Δτ comes from the spacetime interval; it is the moving clock's own elapsed time, smaller than coordinate time Δt by the Lorentz factor.
Proper time is defined only between events a clock can actually pass through (timelike separation); it is the parameter that ticks along a worldline.