The Moon does not emit its own light; we see sunlight reflected from its surface. The changing phases arise because, as the Moon orbits the Earth, different portions of the Moon's sunlit side face our planet. The geometry between the Sun, Earth, and Moon determines how much of the illuminated half is visible to us.
The Moon's cycle of phases (the synodic month) takes about 29.5 days — the time the Moon needs to return to the same phase, as seen from Earth. This is different from the sidereal month (~27.3 days), which measures the Moon's orbit relative to the background stars. The difference comes from Earth's motion around the Sun during the Moon's orbit.
Use a small telescope or binoculars to examine terminator shadows near first and third quarter: the contrast along the line between light and dark highlights mountains and craters. Full Moon is bright and washes out faint objects, but it’s great for spotting larger maria and bright landmarks.
Phases affect tides, night-sky brightness, and observing plans. For deep-sky observing you typically choose darker nights away from the full Moon, while lunar observers may plan sessions around specific phases to capture desired surface detail.
Try this at home: hold a small ball (the Moon) at arm's length and shine a flashlight (the Sun) on it while slowly walking around a chair (the Earth). The changing lit portion will mimic the Moon's phases and make the geometry clear.