How long does it take the Moon to orbit the Earth?

Measured against the stars, one complete lunar orbit takes 27.3217 days: this is the sidereal period, the one used in celestial mechanics calculations. An observer on Earth, however, isn't measuring that. While the Moon travels its orbit, our planet is also moving around the Sun, so the Sun-Earth-Moon geometry takes longer to repeat. Returning to the same phase, from new moon to new moon, takes 29.53 days: the synodic period, the one that sets the calendar of phases and the only one noticeable to the naked eye. The difference between the two isn't a measurement error, it's the result of choosing a fixed reference point versus a moving one.

The Moon takes almost exactly as long to rotate on its own axis as it does to complete its orbit, about 27 days. That coincidence, called synchronous rotation, keeps the same hemisphere facing us: Earth and the Moon are tidally locked, and their rotations are synchronized so closely that we only ever see one side. The opposite hemisphere, the far side, isn't darker than the near side; it simply stays out of view from Earth's surface.

The average distance between Earth and the Moon is 384,400 kilometers (238,855 miles), and the satellite covers it at an average orbital speed of 1.022 km/s. The orbit is elliptical and tilted relative to the plane in which Earth orbits the Sun, so the Moon doesn't repeat exactly the same position on every lap. That's why eclipses, which require an almost perfect alignment of the three bodies, don't happen every month.