In one sense a black moon is the opposite to a blue-moon, its when a new moon occurs for a second time within one month, like a blue moon, when a full moon occurs a second time in one month. The usual would be to have a new and a full moon about once a month.
In another sense, the Black Moon, or Lilith as it is often known, is a hypothetical point in a chart. This point arises due to the Moon's path around the ecliptic forming an axis of the perigee, which is when the moon's orbit is closest to the earth and the apogee, which is when its orbit is furthest from the earth. It is an empty area, a void created, an empty focus. Moving at about 40 degrees a year, the perigee is from where the path of the Dark Moon is calculated. It takes a little under nine years for it to complete its course.
In astrology, the Dark Moon is representative of the darker side of our nature, the negative side. It is the hidden depths that we know exist yet we prefer not to acknowledge. It is subtle on the surface, yet profound beneath. It is the foundations of our personality and the shocking truth that we often deny and project onto others instead. The refusal to see what is really within ourselves. The subtle refusal to see what part we may have played in some unpleasant drama of our lives, how we ourselves have manifested our fears. It is the kind of thing that we intrinsically know we have to deal with, confront the darker side in order to progress unhindered, yet most often we try to ignore. It is the deeper truth that we are forever in search of within ourselves. Probably the best use we can make of the dark moon is to explore our hidden depths, realise that we all have that darker side and go within to resolve that which ultimately hinders us.
I hope that this has answered your question. Its about time someone did, you've waited long enough.