There are a lot of weird moons in our solar system, but ours is probably the weirdest. For one thing, it's really damn big. It's practically a planet.
The Moon is one fourth the size of Earth. With the exception of Pluto and Charon, it is larger in comparison with the planet it orbits than any other satellite in the solar system.
Not only that, but the Moon is so large that it doesn't really orbit Earth at all. Both bodies orbit a point located between them called the barycenter. This is located about a thousand miles below the surface of Earth...but still some three thousand miles from the center of our planet. This makes the motion of the Earth-Moon system as it orbits the sun something like a spinning dumbbell with the weights on the ends mismatched.
Another thing that sets the Moon apart from every other natural satellite in the solar system is the fact that nowhere is its orbit convex to the sun. That is, nowhere is its orbit curved toward the sun. The reason for this is that the gravitational pull of the sun on the Moon is greater than that of Earth.
The mental image most people have of the path the Moon traces as it moves around the sun is a series of loops like this:
or perhaps a wavy line like this:
The fact is that the orbit of the Moon as it moves around the sun is barely distinguishable from a circle. In fact, you have to zoom into a very small section of its orbit just in order to detect the variation. French space artist/astronomer Lucien Rudaux showed what the orbit of the Moon really looks like in this illustration. The lower sketch is to scale:
So it could perhaps rightly be said that the Moon's orbit around the sun is only slightly perturbed by the presence of Earth. In fact, if Earth disappeared, the Moon would continue to orbit the sun pretty much as it does now.
So...why isn't the Moon considered to be a planet? And consequently the Earth-Moon system a double planet?
The main reason for not considering Earth and our moon to be a double planet is that the barycenter lies beneath the surface of Earth. If the Moon were either about a third larger or a third further away, the barycenter would be above the surface of the earth and the two worlds would be a true double planet. And since the Earth-Moon system is not a double planet, the only remaining possible designation for our moon is that it is a satellite of Earth.
So no, the Moon is not officially considered to be a planet in spite of the fact that it has many of the characteristics of a planet. Which only goes to underscore the arbitrary nature of the definition of the word "planet" and how blurry many of the distinctions really are...and why so many astronomers are so unhappy with the current IAU definition of the word.
Why, for instance, are worlds that are as much alike as Mercury, Ceres and Pluto considered to be a planet, an asteroid and a dwarf planet respectively, while worlds as wildly different as the earth and Jupiter are classed together as planets? Perhaps just as we have different categories of planet—dwarf, minor, terrestrial, gas giant, etc.—we need one more to include special cases like our moon.