Why does the Moon change shape?

The Moon looks like a crescent, a half-disc and a full circle during the month. Its shape stays the same. Our view of its sunlit half changes.

▶ Try it yourself: move the Moon around Earth and watch the visible lit fraction.

How does the phase follow the orbit?

If ε is the angle between the Sun and Moon as seen from Earth, the visible illuminated fraction is (1 − cos ε) / 2. It is 0 at new Moon, 0.5 at either quarter and 1 at full Moon.

The phase cycle is the 29.53-day synodic month, from new Moon to new Moon. It is longer than the Moon’s orbit against the stars because Earth also moves around the Sun.

Is the dark part Earth’s shadow?

No. Earth’s shadow points directly away from the Sun. A lunar eclipse happens only when the full Moon also crosses that shadow. The Moon’s tilted orbit keeps that from happening every month.

The model leaves out elliptical orbits, libration, changing distance, Earthshine and surface relief in the terminator.

Sources: NASA, Moon Phases; NASA, Eclipses and the Moon; NASA SVS, Moon Phase and Libration.