Turns out, it orbits the Earth, despite these claims. The above claims are true, but are not important in this argument. Instead, you have to look at something called the Hill sphere. Basically, it’s the volume of space around an object where the gravity of that object dominates over the gravity of a more massive but distant object around which the first object orbits.
OK, in English — and more pertinent to this issue — it’s the volume of space around the Earth where the Earth’s gravity is more important than the Sun’s. If something is orbiting the Earth inside Earth’s Hill’s sphere, it’ll be a satellite of the Earth and not the Sun.
The derivation of the math isn’t terribly important here (and it’s on the Wikipedia page if you’re curious), but when you plug in the numbers, you find the Earth’s Hill sphere has a radius of about 1.5 million kilometers. The Moon’s orbital radius of 400,000 km keeps it well within the Earth’s Hill sphere, so there you have it. The Moon orbits the Earth more than it orbits the Sun. In reality it does both, and saying it orbits one and not the other is silly anyway.
(image credit: NASA)