... like I'm 5 years old
The Moon does not actually change shape. Sunlight always lights roughly half of it, but as the Moon travels around Earth, we see different amounts of that sunlit half. Those changing views are called phases. The Moon is still a whole, round world when it looks like a thin crescent—or disappears from view at new moon.
Imagine noticing a slim crescent above the rooftops after sunset. Over the following evenings, the bright part grows. This is a waxing Moon. It becomes a half-lit disk, then a fuller-looking one, until nearly all the side facing us is bright. After the full Moon, the visible bright part shrinks: the Moon is waning. The pattern then begins again.
It is tempting to think Earth’s shadow covers the missing part. Ordinarily, it does not. The dark portion is part of the Moon that direct sunlight is not lighting from our point of view. Earth’s shadow falls on the Moon during a lunar eclipse, a separate event that does not happen every month.
You can follow the change yourself without any equipment. Find the Moon on a few clear evenings and notice where it appears, which part is bright, and whether that part is growing or shrinking. If you miss it one evening, look in the daytime or early morning: the Moon is not confined to the night sky.
Think of walking around a lamp-lit basketball. The ball stays round, and the lamp keeps lighting one side; what changes is how much of that bright side you can see.
... like I'm in College
Start with the Sun, Earth, and Moon. The Sun lights the lunar hemisphere facing it. Meanwhile, the Moon orbits Earth, so our viewing direction changes relative to that sunlit hemisphere. At new moon, the Moon is roughly between Earth and the Sun, and its illuminated side faces mostly away from us. At full moon, Earth is roughly between the Sun and Moon, and we see nearly all of the illuminated side.
Between those points, the Moon presents crescents, half-lit disks, and gibbous phases—disks that look more than half lit. The names first quarter and third quarter describe positions in the phase cycle, not how much of the visible disk is bright. At either quarter phase, roughly half the disk we see is illuminated. A complete cycle from one new moon to the next takes about 29.5 days.
The phase also offers a clue about when to look. A first-quarter Moon is often well placed in the evening; a full Moon rises around sunset; a third-quarter Moon is often visible in the morning. These are useful guides, not promises about an exact clock time at every location.
Why no eclipse at every new or full moon? The Moon’s orbit is tilted relative to Earth’s path around the Sun, so the three bodies usually do not line up precisely enough for one to cast a shadow on another. Their near-alignment can still matter on Earth: it helps explain the pattern of high and low tides. For a closer look at the changing view itself, NASA’s guide to Moon phases follows the cycle across the sky.
Imagine clearing a table for a tiny solar system. Build a yellow Sun at one end, a blue Earth near the middle, and a rounded gray Moon that you can move around Earth. Put a small mark on the Moon’s Earth-facing side. The mark will help you keep track of your viewpoint as the model moves.
Now place a lamp near the yellow bricks. The bricks label the Sun; the lamp supplies its light. Turn down other lights, and look from beside your Lego Earth rather than from above the table. Wherever you put the Moon, notice which part the lamp illuminates and how much of that lit part you can see.
Move the Moon to the side of Earth nearest the lamp. From Earth’s position, you look toward mostly unlit lunar bricks: new moon. Slide the Moon a quarter of the way around. About half of the side you see is bright: first quarter. Move it to the far side of Earth, where its lit face turns toward you: full moon. Another quarter-turn brings a half-lit third quarter. The positions between them give you crescents and gibbous Moons.
There is one important adjustment. If your Lego Earth blocks the lamp at the full-moon position, lift the Moon slightly so light reaches it. In space, the Moon’s tilted orbit usually carries it above or below Earth’s shadow. Leave it in the shadow only when you want to model a lunar eclipse.
Nothing has been removed from your Lego Moon. You have moved it, changed your view of its lighting, and recreated the reason its familiar shape seems to change.
... like I'm an expert
The visible phase is a question of Sun–Moon–observer geometry. Define the phase angle, α, at the Moon: it is the angle between the directions toward the Sun and Earth. In a simple model of a spherical Moon lit by a distant Sun, the illuminated fraction of its apparent disk is (1 + cos α) / 2. At α near 0°, the disk appears full; near 90°, half-lit; near 180°, new. The curved boundary between lunar day and night is the terminator.
That model describes illumination, not the Moon’s changing physical outline. Nor does it require Earth’s shadow. An eclipse needs a more particular alignment, including a crossing near the plane of Earth’s orbit. The Moon’s orbital inclination is about 5°, which is why most new and full moons pass without an eclipse. NASA’s eclipse geometry explanation distinguishes those shadow events from ordinary phases.
Two different “months” are relevant. The Moon takes about 27.3 days to orbit Earth relative to distant stars, but about 29.5 days to return to the same phase. During one lunar orbit, Earth also advances around the Sun; the Moon must travel farther before the Sun–Earth–Moon viewing geometry repeats.
The Moon rotates once per orbit, keeping approximately the same face toward Earth. Small changes in our view, called libration, let us glimpse a little around its edges over time. Libration does not cause the phases: the dominant change is still how much of the sunlit hemisphere faces us. On a clear crescent night, you may also discern a faintly glowing remainder of the disk. That is earthshine—sunlight reflected from Earth onto the Moon and back toward us.