In more detail
The cycle from one new moon to the next is the synodic month, about 29.53 days, longer than the 27.32 days the Moon takes to orbit relative to the stars because the Earth has moved along its own orbit meanwhile. The eight named phases are new, waxing crescent, first quarter (about 7.4 days in), waxing gibbous, full (about 14.8 days), waning gibbous, last quarter and waning crescent. The lit fraction is roughly (1 − cos(2π × age / 29.53)) / 2. The Moon is full when it is opposite the Sun in the sky and new when it is in the same direction; from the southern hemisphere the lit side appears reversed.
Why it matters
Tides are largest (spring tides) around new and full moon. Phase matters for astronomy and night photography, where a bright Moon washes out faint stars, and for calendars such as the Islamic months and the date of Easter. Simple calculators use the mean cycle, but because the Moon's orbit is elliptical the true moment of a phase can differ from the mean by up to about 14 hours.
Example
Reference new moon: 2000-01-06 18:14 UTC
Days to 2026-10-09 00:00 UTC: 9,772.24
9,772.24 / 29.530589 = 330.91925 cycles
0.91925 × 29.530589 = 27.15 days old → waning crescent, about 6 % lit
The same method puts the new moon of the 12 August 2026 total solar eclipse at day 28.95 of a cycle, about 14 hours from the mean value, which shows the size of that error.