Proper time is the time actually experienced by a specific clock or object as it moves along its own worldline, as opposed to the time measured by some outside observer. It is the reading on a wristwatch strapped to the object itself.
Two objects that start and end at the same two events in spacetime can experience different amounts of proper time if they took different paths to get there, this is the basis of the famous twin paradox. A clock moving through a stronger gravitational field, or moving faster relative to another observer, ticks through less proper time.
Because proper time depends on the path taken, it is a more fundamental quantity in relativity than the ordinary, frame-dependent notion of time most people are used to.
One of the clearest real world confirmations of proper time comes from cosmic ray muons. These particles are created high in Earth’s atmosphere and, according to their own extremely short natural lifetime, should decay long before reaching the ground, yet detectors on the surface catch them constantly. From our frame, their fast motion through spacetime slows their clock relative to ours, stretching how much proper time we observe passing for them and letting far more of them survive the trip down than a naive calculation would predict. GPS satellites face a related correction, their onboard clocks run faster than clocks on the ground due to weaker gravity at orbital altitude, and without adjusting for that mismatch in accumulated proper time, GPS positioning would drift by several kilometers a day.
The twin paradox makes the effect vivid: if one twin stays on Earth while the other takes a high speed round trip to a distant star, the traveling twin returns having aged less, having accumulated less proper time along their particular path through spacetime, even though both twins experienced time passing normally from their own perspective the entire way.