The idea of the quarters of the Moon always used to confuse me. I couldn't quite see what looked like a half moon as the first quarter. Instead of considering the shapes that we see, we have to think of the whole cycle of phases divided into four quarters. It begins with the new Moon, and a quarter of the way through you have the waxing (growing) half Moon. A second quarter gives you a full Moon. Etc.
But it's still confusing, because these days we usually only use this terminology for the first and last quarters.
1. At the end of the first quarter, you end up with a half-illuminated Moon.
2. At the end of the second quarter you have a full Moon.
3. At the end of the third quarter it's another half Moon, but it's on the opposite side to the first quarter.
There's still another week to go until the end of the cycle, so I don't know how the third quarter also gets labeled as the last quarter. Mysteries abound.