Why do we always see the same side of the Moon?
The Moon also rotates about its own axis just as the earth is doing. But it does so much slower. One rotation actually takes the same time as it takes the Moon the complete one orbit around the Earth. And the Moon orbits the Earth in the same direction as the Earth is rotating (towards the east if you want to call it that), and the Moon rotates about its own axis again in the same direction. All this means that we are always looking at the same side of the Moon.
That is no accident and happens quite often with spherical Moons. The gravitational pull between Earth and Moon causes the Moon (and the Earth to a lesser extent) to bulge out in the direction of the Earth as well as on the opposite side of the Moon. The solid Moon resists this deformation and tends to make it constant, in the sense that the bulge always stays at the same position in the Moon’s body. It can do so by adjusting its own rotation to become synchronous with its orbital period about the Earth. This is called tidal locking.
In this animation the closer moon is in tidal locking with the central planet. The moon that is further out has its own independent rotation.
Thus both the rotation of the Moon about its own axis (the "Moon day") and its revolution around Earth (Lunar cycle) take the same time, called the Synodic month of 29.53 days.