The following descriptive definition comes from An Introduction to Chaotic Dynamical Systems by Robert Devaney. This is most easily explained by considering the construction given below. The complement is of course all the points not in the set. A set is totally disconnected if it contains no intervals. In particular if you choose any two points in the set there is always space between them.
Every point has some elbow room on both sides.
A set is perfect if every point in it is an accumulation point or limit point of other points in the set. This property is actually redundant since the set being closed implies this property. It is a good reminder, however, that although the points have space around them, they all have neighbors living aribtrarily close to them. They are spread out, but no one is alone. Note that because this definition is purely topological it is not limited by the dimension of the metric space in which the sets are constructed.