Current Phase of the Moon:

 

 

So why does the Moon go through phases?

Unlike the Sun, the Moon does not give off its own light -- we can only see that part of the Moon that is reflecting Sunlight. As the Moon orbits the Earth, we see a different amount of that reflected Sunlight, as shown below:

Contrary to popular belief, there is no "dark side of the Moon." The Moon rotates on its axis (a "lunar day") in the same amount of time it takes it to go around the Earth (a "lunar year"). Therefore, one lunar day equals one lunar year, and the same side of the Moon is always facing us.  So, during a New Phase of the Moon, the back side that we never see is illuminated by the Sun!  (See diagram).
    Sometimes during the Full Phase of the Moon, Earth will cast a shadow on the Moon, producing a  Lunar Eclipse:

Similarly, during the New Phase of the Moon, the Moon may block out the Sun (it is only a coincidence that both the Sun and the Moon appear to have the same angular diameter).  This is a Solar Eclipse, and all or part of the Sun may be blocked by the Moon:

So why don't we have a lunar and solar eclipse every month?  The orbit of the Moon around the Earth is tilted about five degrees off the plane described by the Earth orbiting the Sun.  So, more often than not, the Moon is above or below the Earth-Sun plane -- during a full Moon, the shadow cast by Earth could extend into space above or below the Moon, and during a new Moon, the Moon is above or below the Sun.  That is what makes eclipses so special -- they only happen a few times a year!