Why we only ever see one side of the Moon
The Moon does spin. It just turns exactly once for every trip around Earth, a state called tidal locking.

Photo: NASA/JPL/USGS / Wikimedia Commons, Public domain
Look up at the Moon on any night and you see the same familiar pattern of dark plains and bright highlands. That is not because the Moon stands still. It rotates once on its axis for every orbit around Earth, roughly every 27.3 days, so the same face always points toward us.
How tidal locking happens
Earth's gravity pulls slightly harder on the near side of the Moon than on the far side, stretching it into a gentle bulge. Billions of years ago the young Moon spun faster. As it turned, Earth's pull on that bulge acted like a brake, slowing the spin until rotation and orbit matched. Once they matched, the brake stopped working against it, and the Moon stayed locked.
Tidal locking is common. Many moons in the solar system, including the large moons of Jupiter, keep one face toward their planet.
We see a bit more than half
Because the Moon's orbit is slightly oval and tilted, it appears to rock and nod a little over a month. This wobble, called libration, lets observers on Earth see about 59 percent of the surface over time.
The far side is not dark
The side we never see is often called the dark side, but it gets as much sunlight as the near side. It was first photographed in 1959 by the Soviet probe Luna 3. In January 2019, China's Chang'e 4 made the first soft landing on the far side.
Earth's pull also works the other way: the Moon's tides slowly lengthen our day and push the Moon farther away, by a few centimetres a year.

