In observational astronomy classes, one of the first things students learn about are how the phases of the Moon work.
Notice how a new moon exists when the Moon is between the Sun and the Earth such that its far side (the side we never see from Earth) is lit while the side facing us is dark. It's also overhead during the day so we simply don't see it.
Next we have a waxing crescent, first quarter moon, waxing gibbous, and full moon. Note that the right side of the Moon, as seen from the Earth, is getting progressively more and more lit as you advance towards the full moon.
The full moon occurs when the Moon is on the opposite side of the Earth from the Sun so the side facing us is now fully lit. The Moon's also high overhead in the middle of the night making it easy to spot in the sky.
From the full moon, we then see a waning gibbous, third quarter moon, waning crescent, and finally the new moon returns. During the waxing phase of the lunar cycle, the left side of the moon is lit less and less with each passing day.
This cycle, from new moon to new moon, takes 29.53 days (29 days, 12 hours, and 44 minutes). Note that the phases are simply due to the geometric orientation of the Moon, Earth, and Sun and the simple fact that the half of the Moon facing the Sun is lit and the half facing away from the Sun is dark (just like the Earth). As the Moon orbits the Earth, we see the lit and unlit portions at different angles. Contrary to popular belief, the phases of the Moon have nothing to do with the shadow of the Earth on the Moon (lunar eclipses do, but that's a different story).
One thing that students often have a hard time remembering is which half of the Moon is lit when it's waxing (right side) and which side is lit when it's waning (left side). One easy way to do so, at least for me, is to remember the symbol shown below.
This is a Wiccan symbol representing the goddess (who is associated with the Moon). The left side represents the waxing crescent moon (right side of the Moon lit), the middle circle represents the full moon, and the right side represents the waning crescent moon (left side of the Moon lit). The symbol is sometimes called the Triple Goddess since it represents her three aspects as maiden, mother, and crone (waxing, full, and waning).
While scholars debate about the origin of this symbol and the religious beliefs of ancient European cultures, we can say with certainty that ancient cultures around the world had an intimate familiarity with the cycle of lunar phases - it was, after all, how you kept track of time before calendars were invented. Remember this symbol and you'll instantly know, when looking at a crescent, quarter, or gibbous moon, whether it's waxing or waning in its neverending cycle of phases.