propagator
The propagator is the amplitude for a particle to travel from one place at one time to another place at a later time. If you ask, 'given that the particle was here, what is the amplitude to find it over there after a certain interval?', the propagator is your answer. Squaring it gives the probability density, so it is the working tool for predicting where a quantum object is likely to turn up.
It is the position-space face of the time-evolution operator. Multiplying an initial wavefunction by the propagator and adding up the contributions from every starting point delivers the wavefunction at the later moment. In this sense the propagator is a kernel: it smears the old wavefunction across space in just the way the dynamics demands, weaving the future shape out of the present one.
Richard Feynman gave the propagator its most vivid meaning. He showed it can be computed by summing a phase factor over every conceivable path the particle might take between the two points — not just the classical straight line, but every wiggling detour. The paths interfere, and out of that interference the familiar quantum behaviour emerges. The propagator is where the path-integral picture of quantum mechanics comes to life.
The propagator K carries amplitude from x₀ to x; integrating it over all starting points evolves the wavefunction.
The propagator gives a probability amplitude, not a probability. As with the wavefunction itself, you must take the modulus squared before you have anything you can interpret as a chance of finding the particle.