A "new moon" is considered the first phase. This is when there appears to be no moon in the sky, because the moon is between the earth and sun. Hence the illuminated half of the moon is facing away from our view on earth.
"Waxing crescent" is the lunar phase that immediately follows a new moon. The phase is easier to understand once the words are defined. "Waxing" refers to the moon expanding, and "crescent" is when the moon is less than a quarter full, or illuminated. So waxing crescent is the phase between a new moon and first quarter, when the moon's light is growing but less than a quarter of the moon is illuminated.
Two "quarters," also called "half moons," occur in the lunar phases. Each quarter is marked when the moon is at a 90-degree angle from the center, with the earth being the center point. The "first quarter" is identified as such because it follows the new moon, whereas the "third quarter" follows the full moon.
When the moon is in the "waxing gibbous" phase, illumination is still expanding. But in this phase, "gibbous" refers to the moon being more than a quarter illuminated.
A "full moon" is when we see the entire half of an illuminated moon. This is when the earth is positioned between the sun and moon.
"Waning" is the opposite of "waxing," in that the moon is losing illumination. In the "waning gibbous" phase, the moon is more than a quarter illuminated, but as the phases progress, the illumination will become less and less.
"Third quarter" moon looks the same as the first quarter moon, in that the moon is situated at a 90-degree angle to the earth and sun, but this phase occurs as the moon is losing illumination.
"Waning crescent" is the last lunar phase before the next new moon. The moon is closer to the sun at this phase and will continue to lose illumination, hence the waning characteristic. It is also in the crescent phase because less than a quarter of the moon is illuminated.