If something's dichotomous, it's divided into two distinct parts. It can describe a plant whose leaves pair off in opposing buds or anything — a government, a relationship — that has two divisions that are sharply opposed.
The relationship between science and mysticism, or astrology and astronomy, for example, are two examples of a dichotomous relationship: each side has a completely irreconcilable set of basic principles. You might say the same about a father and son who support different baseball teams. Not surprisingly, the word dichotomous comes from the ancient Greek word dikhotomia, meaning "a cutting in half." So be warned and make sure your children grow up supporting the same team as you.