An easy way to understand the moon cycle is 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 and 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 our 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 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.