Like I said: I should have known this, but I didn't.
You have probably seen the moon. If you have seen the moon, you may have noticed that its apparent shape in the night sky changes on a daily basis. If you are a REAL smarty pants you may have even heard words like "waxing" or "waning" or "crescent" or "gibbous." These are good words. Important words. A waxing moon is increasing in size, and a waning moon is decreasing in size. A crescent moon has a concave curve, while a gibbous moon has a convex curve. These are things that people know, for the most part. And many people also understand WHY these shapes occur... but I did not. I'm not too proud to admit it. I always assumed that the phase of the moon, that is the "fullness" of the moon, was simply determined by the shadow of the earth on the moon...but this explanation wasn't satisfying to me because I couldn't figure out how the earth, a spherical object, could cast a shadow on the moon that was anything other than round. How could the shadow of the earth create a gibbous moon, a moon that is more than half full, with a CONCAVE shadow?
It didn't add up. But luckily there are resources on the internet that can explain this phenomenon in a single figure, such as the one presented below from the website www.moonconnection.com.

So really the earth's shadow has nothing to do with it (except in the case of a lunar eclipse)! Half of the moon is always illuminated by the sun...what changes is our perspective on the moon as it orbits the earth!!! The inner circle of moons shows how the moon is ACTUALLY illuminated over the course of a cycle, and the dotted radii in the diagram show our changing perspective over the course of a lunar cycle. This changing perspective is responsible for the shapes we can see in the diagram's outer circle of moons. And finally, finally, finally I understand how it is possible to have a gibbous moon. But I still have much to learn about how to walk on the moon: