what is the difference between planets and moons, and how do we decide which moons belong to which planet.?

Planets orbit stars. Moons orbit planets. And you can tell which moon belongs to which planet by observing which planet that moon orbits.

Planets orbit stars.
Moons orbit planets.
A moon belongs to the planet it's in orbit about.

A planet orbits the sun.
Moons orbit a planet.

Its easy to know which moons belong to which planet because the moons don't wander away from their planet, they are in orbit.

a PLANET orbits around a SUN.
a MOON orbits around a PLANET.
A MOON belongs to the planet it is orbiting around.

Moons orbit a planet. Planets, in our solar system, orbit the sun.