Relative to the Earth, the Moon makes one rotation every 29.5 days. That happens to also be the time it takes for the Moon to complete one revolution around the Earth. This might seem like a coincidence, but it's not.
In the past, the Moon used to rotate much faster than it does now. But over millions of years, the effect of the Earth's gravity has slowed down the Moon's rotation until it became gravitationally locked to the Earth. This is why we always see the same side of the Moon.
It would seem logical to say that at any one time we can see 50% of the Moon's face. If the Moon were flat, that would be correct, however we know the Moon is a sphere. And the spherical shape of the Moon hides the area close to the perimeter and we can, at any one time, see only 41% of the Moon's face.
Even though the same side of the Moon's faces us, we do see a bit more than half of the Moon's face. Over time, because of librations, we can see up to 59% of the Moon's surface.
Librations are irregular motions of the Moon in its elliptical orbit around the Earth. They are measured using longitudinal and latitudinal coordinates. Both are computed from a central point that is at a fixed geographical location on the lunar surface. This point is in the Sinus Medii, a small plain just below and to the right of he large rayed crater Copernicus. Two meridians emanate from this point: the Central Meridian runs north and south, and the Lunar Equator runs east and west.
Different librations affect different sides of the Moon and each contribute added surface area. The added viewable surface area changes from day to day and month to month. Some librations overlap, but the maximum amount of lunar surface that we can see over time is 59%.
The longitudinal libration is produced by the elliptical orbit of the Moon. Because of the nature of the elliptical orbit, the speed of the Moon changes depending on which part of the orbit it is in. When moving from its fastest point (closest to Earth) to its slowest point (farthest from Earth,) the Moon's speed is slowing down. But, because the Moon's rotational speed stays the same, for a period of time the Moon's face is not pointed directly at us, and this "lag" effect allows observers to see an extra bit of the lunar surface. In effect, we are "peeking" around the edge of the Moon! When the Moon is one quarter of the way around the Earth, it is 97 degrees through its rotation.
This libration is called longitudinal because the extra surface areas exposed are along the lines of longitude (perpendicular to the equator.) The total extra surface area we can see from longitudinal librations is about eight degrees.
There is also a libration that is latitudinal. The plane of the Moon's orbit is titled 5 degrees to the ecliptic. For half the lunar cycle, the Moon is above the ecliptic and for the other half it is below the ecliptic. Each of these half cycles expose an extra bit of the Moon. The extra surface area shown during these librations is at the top of the northern hemisphere or the bottom of the southern hemisphere. The total extra surface area we can see from latitudinal librations is almost 7 degrees.
There is one more librations called Diurnal Librations. This librations occurs every day. Observers can "see" over the top of the Moon as it is rising. Likewise they can see under the bottom when it is setting. Diurnal librations are caused because the radius of the Earth adds an extra 4000 miles of height for looking "over" or "under" the Moon when it is on the horizon. This daily libration gives us an extra one percent of lunar surface area for viewing.