During the crescent phases of the moon (and during new moon in a solar eclipse) it is often possible to still see the the part of the moon that is shadowed from the suns direct rays because light which comes to the earth can be reflected off of the Earth's surface and atmosphere and weakly illuminate this part of the moon. This illumination of the shadowed part of the moon by light reflected off of the earth is "Earthshine". To see how Earthshine occurs due to a double reflection of sunlight. consider the figure above: Light from the Sun is reflected off of the Earth (point A) onto the Moon (point B). Some of this light is then reflected off of the Moon back towards the Earth (point C). Because of this we see part of the Moon illuminated by the Sun and the rest of the Moon dimly illuminated by this doubly reflected light (which we call earthshine). Each time light reflects off a surface (like a planet), it gets dimmer because some of the light is absorbed by the reflecting surface. This means that earthshine is dimmer than moonlight because earthshine is sunlight that has been reflected twice and moonlight is sunlight that has been reflected only once (off the surface of the moon). In addition, the reflectivity of the moon (its "albedo") is less than that of the Earth, which makes earthshine even dimmer.
The moonlight of a full moon, for example, is so bright that, even if it were positioned correctly to reflect light back toward Earth, it would completely overpower earthshine. In fact, we primarily see earthshine during the cresent phases of the Moon (as shown in the diagram) because during these phases: