Moon, Earth's natural satellite from it's birth. Even though moon is around earth from thousands of years but unfortunately from earth we have not seen the complete moon. Yes, the moon we see from earth is just the 50-59% . You might have heard "Dark side of Moon" (No I'm talking about the song by Pink Floyd :P, but I do love that song) or "far side of moon", this dark side of moon was invisible to us until 1959 when the soviet's satellite Luna3 captured the first images of the far side of moon. The far side of moon looks completely different than what we see from earth. It is more thicker than the near side. It is a theory that initially there were two moons and they merged together forming one thicker side and other thinner, but that is for some other article.
The Near side of Moon we see from Earth
The Dark side of Moon
1st image of Dark side of Moon by Luna 3
Anyways, how it is possible to see just one side? It's just Earth's gravity plays an important role in this. Just like earth moon does rotate around it's own axis but the time period required for moon to complete one orbit around earth(Lunar Month) and itself is nearly same 27.3 days. When moon was formed after 30 million years of formation of Earth, it was spinning around its axis at a faster speed than now. The Earth's gravity has slowed down the speed of the moon over a period of time.
Between the line on a moon and earth, the earth's gravity has induced a bulge on the moon. Which makes it a little bit oval. When moon is orbiting around Earth the this bulged part keep moving away from the center line but Earth's gravity pulls it back to it's original Earth-moon line. This caused a torque action resulting into slowing down the speed of moon until it synchronized with the orbit also known as Tidal locking of the moon. Because of this Tidal locking we see only one side of the moon.
You can understand by the visual demonstration from this video. Check out their "Minute Earth" channel if you are interested in this kind of stuff.