A new moon generally can't be seen, except during a solar eclipse. They cross the sky with the sun during the day. The moon's shadow side is toward Earth.
The full moon, opposite the sun in its orbit, shows us its sunlit side. The moon appears full a couple of days before and after the exact moment of full moon.
The first quarter moon is halfway between new and full moon. Half of the moon is illuminated and it's a quarter of the way through it's orbit around the Earth.
As the moon orbits Earth, we see varying fractions of its lighted face. These are the changing phases of the moon. Read four tips to understanding moon phases.
A waxing gibbous moon is in the sky before it's dark and lights up the early evening. It's more than half lighted, but not quite full. It sets after midnight.