It doesn’t matter when you look at the moon you will always see the same side, day or night. This is fairly common, most moons and even some planets show this behaviour. It’s called tidal locking and is the result of the planets pull on the moon. Our moon rotates once on its axis each time it orbits the Earth. If it rotated more than once or less than once then we would be able to see the other side of the moon. But this isn’t just a coincidence, when the moon originally came into our orbit (details unknown) it wasn’t in sync with the Earth as it is now. The moon and the Earth are fairly close meaning that the Earth has a prominent effect on the moon and has gradually adjusted its orbit and shape to how it looks today.
Lots of people refer to the side not facing us as the ‘dark’ side of the moon. This makes it sound like one side of the moon never sees sunlight and this isn’t true. As you can see from the animation above the moon orbits the Earth in the opposite direction to the Earth orbiting the sun. This means it sees plenty of sunlight on all sides.
The moon doesn’t orbit the Earth in an exact circle like the animation above therefore we do sometimes see a sliver of the other side of the moon but nothing spectacular.