It’s probably easiest to understand the moon cycle in this order: new moon and full moon, first quarter and third quarter, and the phases in between.
The new moon occurs when the moon is positioned between the earth and sun. The three objects are in approximate alignment. The entire illuminated portion of the moon is on the back side of the moon, the half that we cannot see.
At a full moon, the earth, moon, and sun are in approximate alignment, just as the new moon, but the moon is on the opposite side of the earth, so the entire sunlit part of the moon is facing us. The shadowed portion is entirely hidden from view.
The first quarter and third quarter moons (both often called a “half moon“), happen when the moon is at a 90 degree angle with respect to the earth and sun. So we are seeing exactly half of the moon illuminated and half in shadow.
Once you understand those four key moon phases, the phases between should be fairly easy to visualize, as the illuminated portion gradually transitions between them.
An easy way to remember and understand those “between” lunar phase names is by breaking out and defining 4 words: crescent, gibbous, waxing, and waning. The word crescent refers to the phases where the moon is less than half illuminated. The word gibbous refers to phases where the moon is more than half illuminated. Waxing essentially means “growing” or expanding in illumination, and waning means “shrinking” or decreasing in illumination.
Thus you can simply combine the two words to create the phase name, as follows:
After the new moon, the sunlit portion is increasing, but less than half, so it is waxing crescent. After the first quarter, the sunlit portion is still increasing, but now it is more than half, so it is waxing gibbous. After the full moon (maximum illumination), the light continually decreases. So the waning gibbous phase occurs next. Following the third quarter is the waning crescent, which wanes until the light is completely gone — a new moon.