proper time
Every clock measures the time along its own path through spacetime — not some universal cosmic time, but the time it personally experiences. That personal, clock-carried time is proper time. Two travelers who part and reunite generally read different elapsed times; each read the proper time along their own worldline.
Proper time tau is the invariant parameter defined by the interval along a worldline: c^2 dtau^2 = ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2, so dtau = dt sqrt(1 - v^2/c^2) = dt/gamma. Integrating, tau = integral of sqrt(1 - v(t)^2/c^2) dt along the path. Because ds^2 is Lorentz invariant, all observers compute the same proper time for a given worldline between two events — it is a property of the path, not of any frame. It is defined only for time-like (or null, where dtau = 0) worldlines.
Proper time is the natural clock for a moving object and the parameter with respect to which four-velocity and four-acceleration are defined. The famous twin 'paradox' is no paradox in this language: the traveling twin's worldline has a smaller integral of dtau because part of it is not straight — and among time-like worldlines between two events, the straight (inertial) one has the LONGEST proper time, not the shortest (the reverse of the Euclidean triangle inequality, because the metric is indefinite). Caveat: dtau/dt = 1/gamma is the time-dilation factor, but proper time is the invariant while 'dilated time' is the frame-dependent coordinate time.
A muon whose lab lifetime is dilated to gamma times its rest lifetime still decays after one rest lifetime of PROPER time (about 2.2 microseconds). The extra lab time that lets it reach the ground is coordinate time dt = gamma dtau, not extra proper time.
The particle ages by its proper time; the lab sees more coordinate time pass.
Straightest is oldest: the inertial (unaccelerated) worldline between two events maximizes proper time. This 'longest proper time' principle is the special-relativistic seed of the geodesic principle in general relativity.