When the moon is new, we are unable to see it.
The waxing crescent phase is represented by a thin crescent of light on the right in the Northern Hemisphere
The first quarter phase appears to us as a half moon.
Half-moons and full moons are separated by the waxing gibbous phase. Waxing indicates its growth.
During full moons, we may see the Moon fully illuminated.
Between a full and half moon comes the waning gibbous phase. Waning indicates a decrease in size.
The third quarter moon appears to us as a half moon as well. The first quarter moon illuminates the opposing half.
The declining crescent phase appears as a thin crescent of light on the left in the Northern Hemisphere.