Past full, the right side begins to fall into shadow.
At new moon the Sun and Moon share the sky, rising and setting together. The Moon is unlit from our viewpoint and effectively invisible, drowned in the Sun's glare. New moon nights are the darkest of the lunar month, ideal for deep-sky observing and a recurring window for lunar landings, the surface near the terminator is favorable for navigation.
In the days after new moon, the Moon emerges into the western sky just after sunset, a slim crescent thickening night after night. The unlit portion is often faintly visible too, lit by earthshine: sunlight reflected off Earth's clouds and oceans back onto the lunar night side.
One quarter of the way through the cycle, the Moon shows a perfect right-half disc. It rises around noon and is high in the sky at sunset, which makes the terminator (the day-night line) exceptionally sharp through a telescope, every crater along it casts a long shadow.
Between first quarter and full, the Moon is mostly illuminated but still has a noticeable dark edge on the left. It dominates the evening sky and sets after midnight. Shadows are shorter than at quarter phase but still long enough to give lunar features dramatic relief.
At full moon the disc is fully illuminated and rises at sunset, dominates the sky all night, and sets at sunrise. With no shadows along the equator of the disc, surface detail flattens out, but ray systems from young craters like Tycho and Copernicus blaze across the face. Full moons closer to Earth than average are called supermoons.
After full moon the Moon rises later each night, an hour or so later per day on average. The illuminated portion shrinks from the right, and the Moon is best seen in the late evening through early morning hours.
Three quarters of the way through the cycle, the Moon shows a perfect left-half disc. It rises around midnight and is high in the sky at dawn. Astronomers prize this phase for the same reason as first quarter: sharp shadows along the terminator make crater rims and ridge lines pop.
In the final days of the lunar month the Moon becomes a thin crescent visible in the pre-dawn eastern sky, lit on its left limb. Earthshine often illuminates the dark portion. After this the Moon disappears into the Sun's glare and the cycle resets.