The short answer
Phases happen because the Moon orbits Earth. As it moves, we see different fractions of its sunlit side. The Moon is always half-lit by the Sun; what changes is how much of that lit half faces us.
Eight named phases
Astronomers name eight positions in the Moon's orbit: New Moon (Moon between Earth and Sun — dark side faces us), Waxing Crescent, First Quarter (right half lit in the north), Waxing Gibbous, Full Moon (Moon opposite the Sun — fully lit face toward us), Waning Gibbous, Third Quarter, and Waning Crescent before the cycle repeats. The full cycle takes 29.53 days — one synodic month.
The NASA Moon Phases page has an interactive diagram showing the geometry in real time.
Why not a lunar eclipse every month?
Earth's shadow is narrow and the Moon's orbit is tilted about 5° relative to Earth's orbital plane. Most full Moons pass above or below the shadow. A lunar eclipse only occurs when the Moon crosses the orbital plane while full — roughly 2–4 times per year. The NASA Eclipse page lists all upcoming lunar eclipses.
Tidal locking explained
We always see the same face of the Moon because its rotation period matches its orbital period — both roughly 27.3 days. Earth's gravity raised a permanent slight bulge in the Moon's crust early in its history; over billions of years this torque slowed the Moon's spin until it locked. The phenomenon is called synchronous rotation or tidal locking, and it applies to many moons in the Solar System. Read more in our Tides explained guide.
Phases and the calendar
The 29.53-day synodic month is the basis of lunar calendars used by many cultures, including the Islamic Hijri calendar and the Hebrew calendar. Our modern Gregorian calendar months are solar approximations and don't align precisely with Moon phases — which is why the date of Easter (based on the first Sunday after the first full Moon after the spring equinox) moves around each year. The IAU explains the history of astronomical calendars.