How long does it take the moon to orbit the earth?
There are really two answers, either 27.3 days or 29.5 days.
These two different periods are described here. If you draw the elliptical orbit of the moon, and you measure the time it takes to go around the ellipse as it slides through space, you get the siderial orbital period. That is 27.3 days.
But that "sliding through space" is actually following Earth's orbit, which itself curves around the Sun. So when the Moon gets around its elliptical orbit in the siderial cycle, the Sun's light is coming at it from a different angle and we do not see a full cycle of the Moon's phases. To compensate for this effect we must wait a little longer, 29.5 days total, for the Moon to complete a cycle with respect to the Sun-Earth pair. This 29.5 days is the synodic period and it represents the "monthly" cycle we see from one new moon to the next.