Moon Phase
A moon phase is the shape of the sunlit part of the Moon seen from Earth, which cycles from new to full and back over about 29.5 days.
Definition
A moon phase is the shape of the Moon's sunlit half that we can see from Earth, changing as the angle between the Sun, Earth, and Moon shifts while the Moon revolves around us; half of the Moon is always lit, no matter which phase we see. The visible fraction cycles from a thin crescent through a full circle and back over about $29.5$ days, with growing phases called waxing and shrinking ones called waning. That fraction depends on the phase angle $\theta$ between the directions to the Sun and Earth as seen from the Moon, following $(1 + \cos\theta)/2$, and the $29.53$-day synodic month runs longer than the $27.32$-day sidereal orbit because Earth itself advances about $27^\circ$ around the Sun during each lunar orbit, a relation captured by $1/T_{\text{syn}} = 1/T_{\text{sid}} - 1/T_{\text{year}}$. Because the Moon's orbit tilts about $5^\circ$ from the ecliptic, new and full moons usually miss the Sun-Earth line, so eclipses happen only a few times a year instead of every month.
Example
The full cycle runs new moon, waxing crescent, first quarter, waxing gibbous, full moon, waning gibbous, third quarter, and waning crescent, and takes about a month start to finish. At new moon the Moon sits between Earth and the Sun so its lit side faces away from us, and at full moon Earth sits between the Sun and Moon so we see the whole lit side; first quarter arrives about $7.4$ days after new, when the Moon is $90^\circ$ from the Sun and appears exactly half lit. Working the synodic month from the sidereal one gives $1/T_{\text{syn}} = 1/27.32 - 1/365.25 = 0.03660 - 0.00274 = 0.03386$ per day, so $T_{\text{syn}} \approx 29.53$ days, and a full moon turns out to be about $12$ times brighter than a first-quarter moon rather than just twice, because surface roughness shadows the Moon more heavily at large phase angles.
Key Insight
Moon phases have nothing to do with Earth's shadow falling on the Moon; the dark part is simply the Moon's own night side, the half turned away from the Sun at that moment. The Moon takes only $27.3$ days to orbit Earth but $29.5$ days to cycle through its phases, because Earth moves along its own orbit during that time and the Moon has to travel a little farther to line back up with the Sun. The phase also tells you when the Moon will be in the sky: a full moon rises at sunset and sets at sunrise, a first-quarter moon stands highest at sunset, and a new moon is invisible because it sits in the daytime sky right beside the Sun.