What is the difference between the moon’s quarters and a lunar eclipse?

Like the Earth, the Moon is illuminated by the Sun. Always the half facing the Sun is illuminated, and the half facing away from the Sun is dark. There is only one exception to this ‘always’, namely when the Earth comes between the two other celestial bodies and thus intercepts the light of the Sun for the Moon: it is indeed a lunar eclipse.

The phases of the Moon are due to the movement of the Moon around the Earth. When the Moon is closer to the Sun than we are, we mainly see the back of the Moon that is not lit by the Sun, and the illuminated part is a rather small crescent. From the moment the Moon has moved into its orbit further than we have from the Sun, we see more than half of the illuminated side, and we see all of it when it becomes Full Moon. The quarters are therefore the moments when the Moon is as far from the Sun as the Earth: we can see just half of both the illuminated and the dark hemisphere of the Moon and just half not.

At First and Last Quarter, the Earth is not at all between the Sun and the Moon. You can clearly see that if you look at a quarter of an hour where the Sun and Moon are in the sky.