Definition
Moon phases are the changing shapes of the Moon's bright part as seen from Earth, which happen as the Moon orbits Earth and reflects sunlight. The Moon itself never changes shape. What changes is the angle from which we look at its lit side as it circles Earth.
The Sunlit Half and Our Viewing Angle
Imagine a round ball in the middle of a dark room, with a flashlight shining on it from one side. Only the half facing the light is bright. The other half stays dark. The Moon floating in space works exactly the same way.
The half of the Moon facing the Sun is always lit. But we can only look at the Moon from the surface of Earth. As the Moon travels around Earth, the direction we look at its bright side shifts a little every day.
Sometimes we look straight at the lit side. Other times we see the boundary between light and dark at a slant. The Moon isn't changing. What changes is how much of the bright side we can see from Earth.
From the Invisible Moon to the Full Moon
The Moon's changes follow an orderly pattern set by the angle between the Sun, the Moon, and Earth. When the Moon sits between the Sun and Earth, its lit side faces the Sun and the side facing us is in shadow. We can't see the Moon at all. This is called the new moon. By the way, the Moon always shows the same face to Earth, so the side we see is always the Moon's near side, whatever the phase.
As the Moon moves a little each day from west to east around Earth, a thin sliver shines on the right edge when seen from the Northern Hemisphere, like Korea. This is the waxing crescent. (In the Southern Hemisphere, left and right are flipped.) A few days later, the right half is fully lit. This is the first quarter moon, and it's easy to spot in the southern sky around sunset.
When the Moon reaches the far side of Earth from the Sun and we see its entire lit face, we get the round full moon. After that, only the left half stays lit (the last quarter moon), then just a thin sliver on the left (the waning crescent), and then we're back to the new moon. The whole cycle takes about 29.5 days.
A Closer Look: It's Not Earth's Shadow
Many people think the Moon's shape changes because Earth casts a shadow on it. But Earth's shadow only covers the Moon during a lunar eclipse, a special event that happens just a few times a year. In an eclipse, the Moon slips into Earth's shadow.
Ordinary phase changes have nothing to do with shadows. They happen because the angle from which we see the sunlit half of the Moon keeps changing. It's like holding an apple at arm's length by a window and slowly turning your body in place. The lit part of the apple you can see keeps changing.
To be more precise, the Moon's true orbital period around Earth, the sidereal month, is about 27.3 days. But while the Moon circles Earth, Earth is also moving around the Sun. So the time it takes for the same phase to come back, the synodic month, is about two days longer, at 29.5 days.
Common misconceptions
The Moon changes shape every day because Earth's shadow blocks part of it.
Earth's shadow covering the Moon is a special event called a lunar eclipse. Ordinary phase changes happen because we view the sunlit side of the Moon from a different angle.
Where you meet it
In a dark room with a round lamp on, hold a ball and slowly turn in place. The bright part of the ball goes from a crescent shape to a full circle.
The traditional Korean lunar calendar is built on the Moon's phases. Day 1 is the new moon, when the Moon can't be seen, and around day 15 the full moon rises. (The exact moment of fullest moon usually falls around day 15 or 16, and occasionally on day 14 or 17.)
In one sentence
Moon phases aren't caused by the Moon changing or by shadows. They happen because, as the Moon orbits Earth, we see its sunlit side from a different angle.