Which planet does the Moon orbit?

The Moon orbits Earth.

Earth’s Moon is bound to our planet by gravity and travels around Earth at an average distance of about 384,400 kilometres. Its path is slightly elliptical, so its distance varies during the month. The Moon completes one orbit relative to the distant stars in roughly 27.3 days; the familiar cycle from one full moon to the next takes about 29.5 days because Earth is simultaneously moving around the Sun.

The Moon does not simply circle Earth while Earth stays still. Both bodies orbit their shared centre of mass, called the barycentre, though that point lies inside Earth. The Moon is also tidally locked: it rotates once per orbit, which is why the same hemisphere generally faces Earth. A frequent mistake is confusing the Moon’s orbit around Earth with Earth’s larger orbit around the Sun.