A waning crescent moon is sometimes called an old moon. It’s seen in the east before dawn.
Now the moon has moved nearly entirely around in its orbit of Earth, as measured from one new moon to the next.
Because the moon is nearly on a line with the Earth and sun again, the day hemisphere of the moon is facing mostly away from us once more. We see only a slender fraction of the moon’s day side: a crescent moon.
Each morning before dawn, because the moon is moving eastward in orbit around Earth, the moon appears closer to the sunrise glare. We see less and less of the moon’s day side, and thus the crescent in the east before dawn appears thinner each day.
The moon, as always, is rising in the east day after day. But most people won’t see this moon phase unless they get up early. When the sun comes up, and the sky grows brighter, the waning crescent moon fades. Now the moon is so near the Earth/sun line that the sun’s glare is drowning this slim moon from view.
Still, the waning crescent is up there, nearly all day long, moving ahead of the sun across the sky’s dome. It sets in the west several hours or less before sunset.
As the moon orbits Earth, it changes phase in an orderly way. Follow these links to understand the various phases of the moon.