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Lunar Phase or Moon Cycle
Observing the lunar phase cycle can be done simply by looking up in the sky each night and noticing what portion of the moon is lit. Of course, when it is a new moon you'll not see the moon at all. Technically, a new moon is overhead at noon. However, the reason we can't see a new moon is because the sun is behind it and it is not lighting up any area of the Moon that is visible to us.
Just as the earth travels around the sun in a counterclockwise direction, the moon travels around the earth  in a counterclockwise direction. So, after the moon enters into its cycle following a new moon the portion of the moon that becomes visible to us will be on the portion that is on our right side. When the portion of the moon visible to us is increasing, it is in its waxing phase. When a small portion of the moon on our right side is lit, the moon is in its waxing crescent phase.
As the moon becomes more lit, or more waxing, approximately 3 to 4 days after the first crescent moon the moon will be about half lit up. Ironically, a half lit moon is a quarter moon. This is because the moon is one quarter of its way  through the lunar cycle. Of course, it will be the right half of the moon that will be visible to us.
As the moon becomes more lit, it gets to the point where it is not full but it is sort of oblong with only a small portion on the left side that is not lit. This is a waxing gibbous moon. 
A few nights later the moon will be full. I guess one could call a full moon a half moon because the moon cycle is 50% complete when the moon is full. 
The very next day the moon starts into its waning stage. When the moon becomes oblong again, this time there will be a small  portion on the right side that is not lit. This is the gibbous waning moon.
A few days later, the moon will appear with its left side lit in its right side not lit. This is a three-quarter moon because the moon is three quarters of the way through its cycle. However, when the moon appears like this, it is more commonly referred to as a quarter moon waning.
 As the moon continues toward completing the cycle only a small portion on the left side of the Moon will be lit. When this happens, the moon will be crescent waning. A few days later the moon will not be visible to us again because it has completed its cycle and will once again be in a new moon phase.
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