Why does the moon have phases?

Half the moon is always lit by the sun, exactly as half of Earth is always in daylight. What changes over the month is how much of that lit half faces us. When the moon sits between Earth and the sun, its lit side points away and we see a new moon. When Earth sits between the sun and moon, the lit face points toward us and we see a full moon. The positions in between produce crescents, quarters, and gibbous shapes. The cycle takes about 29.5 days, slightly longer than the moon's 27.3-day orbit, because Earth is also moving around the sun and the moon must travel a little further to return to the same alignment.

The common wrong explanation is that phases are Earth's shadow falling on the moon. That is a lunar eclipse, which happens only occasionally when the alignment is precise enough for Earth's shadow to actually reach the moon, and it turns the moon a dim red rather than producing a crescent. If Earth's shadow caused phases, they would occur far more often and the terminator line would curve differently than it does.

Two details worth noticing next time you look. The moon is visible during the day about as often as at night, because its position relative to the sun determines when it is above the horizon, not whether it is lit. And the same face of the moon always points toward Earth, a result of tidal locking, where Earth's gravity slowed the moon's rotation over billions of years until its rotation period matched its orbital period. That is why there is a far side we never see from the ground, first photographed by a Soviet spacecraft in 1959, and why calling it the dark side is inaccurate, since it receives just as much sunlight as the side we see.