Can we apply Kepler's law of planetary motion to the moons motion around the earth? I have been trying to understand Kepler's law of planetary motion for a while. So I was thinking, if it would even apply to the moon orbiting around the earth. Does the moon too orbit in an ellipse too? Does its orbit have the same properties as the earth's orbit around the sun?
Yes. Kepler's laws apply to orbits of the Earth, and the moon's orbit is elliptical, with an eccentricity of 0.055

Or at least nearly elliptical. In Newton's theory of gravity, if there are two particles in orbit around each other then they will have an elliptical orbit. If there are more than two bodies then the gravity of other bodies will perturb the orbits from exactly elliptical. A more accurate description of the moon's orbit is "Elliptical + perturbations"

The most significant perturbations come from the sun, and from the fact that the Earth is not a perfect sphere. Other planets and tides also have effects on the Moon's orbit.

There are two wiki pages you should read: Orbit of the moon and Lunar Theory