Why the Moon's phase changes
The Sun always illuminates half of the Moon. As the Moon moves around Earth, we see different portions of that illuminated half. Phases arise from the geometry of the Sun, Earth and Moon; Earth's shadow causes a lunar eclipse, not the normal cycle of phases.
The average time from one new moon to the next is about 29.5 days. The exact duration varies slightly between cycles, so the calendar calculates events rather than adding a fixed number of days.
The waxing and waning Moon
Between new moon and full moon, the visible illuminated fraction increases: the Moon is waxing. After full moon, it wanes until the next new moon. First and last quarter correspond to roughly half of the visible disk being illuminated.
The percentage describes the illuminated fraction of the visible disk, not the Moon's 'power'. The crescent's tilt and position in the sky depend on your location and the time. The calendar diagram is illustrative and does not show your local horizon.
When to observe
At the boundary between day and night on the Moon, the terminator, shadows emphasize craters and mountains. At full moon, most of the near side is illuminated almost straight on, reducing the contrast of fine terrain. A bright Moon also makes faint stars and nebulae harder to see.
The phase alone is not enough to plan an observation: check the weather, moonrise and moonset times, altitude above the horizon and light pollution. This calendar does not calculate local visibility.
Calculations and sources
The phase and events are calculated with the Astronomy Engine library. Event dates use Moscow time (UTC+3); the calendar date may differ in other time zones.