The moon is tidally locked, keeping one face toward Earth. That would suggest we can only ever see half of it. In practice we can see about 59%, because the face it presents is not perfectly steady. It rocks slowly, north and south and east and west, over the course of a month. This is libration.
Libration in longitude
The moon spins at a very nearly constant rate, but it does not orbit at a constant rate — its elliptical orbit makes it move faster near perigee and slower near apogee, in accordance with Kepler’s second law.
So the rotation and the orbital motion drift in and out of step during the month. The moon runs slightly ahead of its own spin near perigee and slightly behind near apogee, letting us peek around the eastern and western limbs by up to about 8°.
Libration in latitude
The moon’s spin axis is tilted about 6.7° relative to its orbital plane. As it goes around, we alternately see slightly over its north pole and slightly under its south pole, by up to about 7°. The same effect that gives Earth seasons gives the moon a slow nod.
Diurnal libration
There is a third, smaller contribution that costs nothing but patience. Earth’s radius is not negligible compared with the Earth-moon distance, so as our planet rotates you are carried to a slightly different vantage point. Watching the moon rise and then watching it set puts your eye about 12,700 km apart, and you see marginally around opposite limbs.
Seeing it
Libration is not subtle if you look for it. Photograph the full moon each month for a year and animate the frames: the moon visibly nods and rolls, and features near the limb — Mare Crisium is the easiest — swing in and out of view.
The moon disc on this site has the computed libration applied, shifted according to the real figures for the date you select, so the same wobble shows up as you move through the calendar.