How does the moon reflect light? We can see the moon in the night because it reflects sunlight. But the light is incident on the opposite side of moon with respect to the observer in the night. 
In this case, how does the moon reflect light?
I am not sure if I have articulated my question correctly.
 A: The problem with most of the earth-moon pictures is that they show the Earth and moon very close together - which suggests that the moon is in the earth's shadow for almost half of the time. 
So in the picture linked to above - it looks like  a full moon should be dark.

The real picture is more like this 

A: I think Marek's comment is the correct answer. I will rephrase it here slightly:
I think one of your assumptions is faulty. Look at this picture. Do you still have a question, after seeing this picture?
A: The surface of the moon is rough rather than shiny so it scatters light in all directions.
A: Although this question is answered, I will say this: the orbital plane of moon is not in the same orbital plane of earth, there is about 5 degrees of inclination.
Example, if earth and sun was in the floor of a room, then the moon is on the opposite of sun, but raised from the floor . Now since sun is larger than the earth, the umbra of the shadow cone is diminishing behing the earth, see here: https://celestrak.org/columns/v03n01/fig-1.gif
Usually the moon is also outside the Pen-Umbra zone. That is why the mon gets illuminated, even being at the opposite side of sun (looking from top). Sometimes however, the moon, while in the opposite side of sun wrt earth, stays in the ascending or descending node, that is called a lunar eclipse.
