Half lit, always
The Moon does not shine. It is a grey rock lit by the Sun, and like any ball in sunlight it is always exactly half lit and half dark. What changes through the month is not the Moon but our view of it. As it travels round the Earth we see the lit half from different angles: edge-on at the start, so a thin crescent; face-on two weeks later, so a full disc.
That cycle of phases takes 29.5 days, which is where the month comes from. At new moon the Moon lies between us and the Sun, and its lit side faces away. At full moon it is on the far side of the Earth from the Sun, and its lit side faces us. In between it waxes, growing from crescent to half to gibbous, then wanes back again.
Try it: step through New, First quarter, Full and Last quarter, and watch which side of the Moon is bright each time.
The same face
The Moon spins exactly once for every orbit it makes, so it always shows the Earth the same face. This is not a coincidence: the Earth’s gravity, pulling slightly harder on the near side over billions of years, slowed the Moon’s spin until the two locked together. The far side was unseen by anyone until a spacecraft photographed it in 1959.
Tides
The Moon pulls on the Earth, and pulls hardest on the side nearest to it. The ocean facing the Moon is drawn up into a bulge. The ocean on the far side is pulled least, and is left behind as the solid Earth is drawn away from it, so it bulges too. Two bulges, on opposite sides. As the Earth turns beneath them, each coast passes through both in a day: two high tides and two low tides, a little over twelve hours apart.
The Sun raises tides as well, about half as strong. When Sun and Moon line up, at new and full moon, their bulges add and the tides run highest.
Shadows and eclipses
Reveal the shadows and each body trails a cone of darkness away from the Sun. When the Moon passes through the Earth’s cone, we see a lunar eclipse: the full Moon dims to a dull red. When the Moon’s much smaller cone touches the Earth, people standing in it see a solar eclipse, the Sun blotted out for a few minutes.
If the Moon’s orbit lay flat, this would happen every month. It does not, because the orbit is tilted by about five degrees, and most months the Moon passes above or below the cones. Only when new or full moon falls where the tilted orbit crosses the Earth’s plane do the shadows meet. That is why eclipses come in seasons, a few each year, and why a total solar eclipse at any one place is a once-in-centuries event.
Try it: reveal the shadows and choose Full. The Moon passes above Earth’s shadow and stays bright. Now switch on Eclipse season: the crossing line swings round to point at the Sun, the full Moon slides into the shadow and turns a dull coppery red, and at New the Moon’s small shadow falls on the Earth.