The side of the moon that we can never see from Earth is often called the "dark side" (a) because it sounds cool, and (b) because darkness is a powerful metaphor for the unknown. In fact, the so-called dark side of the moon is not really dark. Luna has day and night on every part of its surface (just like Earth, only Lunar days are 28 Earth days long). During a new moon, for instance, the "dark side" is flooded with light.
The far side of the moon is dark in the metaphorical sense, in that we never see it. To people who think visually (most of us), lack of information seems like darkness.
How is it that there's a part of the moon we never see? Like many moons of planets in our solar system, our moon, Luna, keeps one side to its primary continually (there's a little wobble, so we can really see about 55% of the lunar surface). The moon is "tide locked" (read about tidal locking on wikipedia). This situation develops with all moons and planets over a long time. The Earth's rotation is slowing, too, and will eventually become tide locked to the sun (not for a very, very, very long time, but scientists can measure the tiny lengthening of the day).
Quibble: to be fair, the far side of the moon does receive less light than the rest of the moon, because it doesn't see any of the sunlight reflected from the earth. So technically, it is a bit darker overall. But nearly nobody who uses the phrase "dark side of the moon" is thinking of it in this way.