Quarter moon in a ten cent town.
But seriously, folks, the moon that we see is a disk, one-half of the sphere that is the moon. We never see the whole moon. Ever. So what we see from here on Earth is half of the moon's visible surface lit up to varying degrees of fullness. The lit "half" that we see here is one quarter of the entire moon, so I suppose that could be where the terms "quarter moon" and "first quarter" come from (as well as "third quarter" and "last quarter"). But, according to Wikipedia, it's not: "the term 'quarter' refers to the extent of the moon's cycle around the Earth, not its shape." Clear?
But hey, you might be wondering, what the heck, then, is a "half moon?" Well, darned if it isn't the same thing as a quarter moon (unlike a half-dollar and a quarter-dollar, which are not the same at all). And when we see the full disk lit up, we call it a full moon, even though it's only half of the sphere that we're seeing. Got it?