This question arose naturally once we discovered that our Moon is something of an anomaly -- even compared to those orbiting giant planets like Jupiter and Saturn, Earth's Moon is unusually large. No other known planet has a moon of such comparable size to their parent**.
Some early theories included that Earth's gravity had captured a passing asteroid-like object (highly improbable), or that it had been co-created at the same time as Earth in a kind of binary system (which does not explain why the Moon lacks an iron core as the Earth does).
Upon the discovery that meteor impact craters, instead of the commonly regarded volcanism, were the cause of the Moon's "cheesey" texture, the idea arose: What if it was a large-scale impact knocking into the Earth that created the Moon?
It has been generally regarded as established fact since the 1970s that the Moon was formed in a dramatic impact event. A Mars-sized object came swirling around in the early Solar System and crashed into the new Earth, producing a "ring" of debris that later amalgamated into the single body of the Moon.
However, new evidence is poking some holes in this theory.
Simulations suggest that the Moon should be made out of mostly Theia-like material (non-Earthlike material), but increasingly, studies of lunar geochemistry demonstrate that the elemental isotope ratios of the Moon and Earth are nearly identical.
A group from UC Davis has an alternate theory. Perhaps Earth was impacted with such high energy that the majority of matter from both Earth and Theia was vaporized, creating a hot cloud of thoroughly mixed debris circling Earth's core. The resulting "fat bagel" shape has been dubbed a "synestia". With time, material from the synestia settles into both the Earth and Moon.
It's an intriguing theory, but has been met with skepticism -- after all, ringed planets and protoplanetary disks are common within the Universe, but synestias are theoretical structures that have never been observed. Sarah Stewart, head of this group, has summarized the question of Lunar Origin succinctly:
“The canonical model is in serious crisis. It has not been killed yet, but its current status is that it doesn’t work.”
There seem to be holes in all current models, but that is the march of science -- and it's only a matter of time before an answer can be divined.
(Images are from this lovely article, which is much more in-depth and I highly recommend!)