Ask most people why the Moon changes shape and they say the Earth's shadow covers part of it. It is a natural guess, and it is wrong. The Moon is half lit by the Sun all the time. What changes is how much of that lit half we can see from here.
Drag the Moon, or use the arrow keys when it is focused. Only the side facing the Sun is ever lit.
Picture the Moon as a ball with a lamp shining from one side. The half facing the Sun is bright and the far half is dark, wherever the ball sits. The only question is where you stand when you look at it.
At new moon the Moon sits roughly between us and the Sun. Its lit half faces away from us, so we see the dark half, which is almost invisible against the sky. At full moon it sits on the far side of the Earth from the Sun, and we look straight at the lit half. At the quarters we look at it from the side, and see half of the lit half. That is why a half-lit Moon is called a quarter moon: it is a quarter of the way around its orbit, not a quarter of a disc.
A crescent is what you get when you look at the Moon from nearly the Sun's side. The tips of the crescent always point away from the Sun, which you can check next time you see one at dusk. The same logic means the "dark side of the Moon" is not always dark. The far side, which we never see from Earth, gets just as much sunlight as the near side.
The Earth's shadow does fall on the Moon sometimes, and then we call it a lunar eclipse. It can only happen at full moon, when the Sun, Earth and Moon line up, and it lasts a few hours. Moon phases take weeks and appear every month with no alignment needed. The diagram above is flat, so at full moon the Moon looks as if it sits right in the Earth's shadow. In reality it usually does not.
That angle is real. The Moon's orbit is tilted by about 5 degrees against the plane of the Earth's orbit, so most months the Moon passes just above or just below the shadow. Eclipses need the tilt to line up too, which is why they are rarer than full moons.
The Moon takes about 27.3 days to go once around the Earth, measured against the stars. But the phases take about 29.5 days to repeat. The extra two days come from the Earth itself. While the Moon circles us, we circle the Sun, moving about 29 degrees along our orbit in a month. After 27.3 days the Moon is back in the same place among the stars, but the Sun now lies in a different direction. The Moon must travel a little further to reach the new-moon position between us and the Sun again, and that takes about two more days.
This is also why the Moon rises later each day. Because it creeps eastward around the Earth, it reaches your horizon about 50 minutes later each day on average, with the real gap varying from night to night. Press play and watch the "up about" times above: they slide round the clock as the month goes on. A new moon is up by day, a full moon rises at sunset and sets at sunrise, and a last-quarter moon is the one you see in the morning sky.
The phases are not a change in the Moon. They are a change in our viewpoint, repeating in a cycle of about 29.5 days. The lit half stays lit; the angle between the Sun, the Moon and you decides how much of it you see. Anyone can turn that into a rough clock: a Moon that is a thin crescent in the evening is a few days old, and a Moon still up at sunrise is past full.