There have been several ideas since the 1950's; Lunar Capture; Lunar Fission and 'The Great Collider'.
Lunar Capture theory says that the Moon was an independent body that got too close to the Earth and was gravitationally captured. This theory is no longer seriously considered because the Apollo expeditions to the Moon showed that the composition of the Moon has nothing in common with that of asteroids and other bodies elsewhere in the solar system. It lacks iron, nickel and a multitude of 'mantle-like' compounds. It is rich in material similar to the Earth's crust, but devoid of water and other low-temperature volatiles.
Lunar Fission theory says that the Earth was once rotating so fast that it broke up into two bodies. The problem with this idea, again, is a matter of chemistry. If the Moon originated from terrestrial crustal material, its composition should be very similar to crustal material. It is not. It is rich in titanium, which the Earth is not; it is almost devoid of water locked up in compounds which the Earth's crust is very abundant in.
The best modern idea, announced in the 1970's, was that a Mars-sized body slammed into the Earth and ejected a plume of matter into orbit that quickly formed the Moon. Supercomputer calculations show that this material will be chemically different from Earth crustal material because it was heated to high temperatures which would have driven off the water and other volatiles. The heavier mantle material would have quickly settled back into the Earth leaving the Moon without these materials. The Earth and Moon would have been remelted from the heat, so this all must have happened before the Earth was 4 billion years old, but after its initial formation 4.5 billion years ago.