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.