The presence of two tails is a sign that the comet is getting closer to the sun (ISON will be closest to the sun on November 28th, just a few days away).
The prominent tail is called the ion tail. It is composed of ionized gas molecules created by solar wind (charged particles that move very quickly from the sun). This tail points almost directly away from the sun, and has nothing to do with the direction the comet is traveling. As the comet moves away from the sun, the tail leads the way. Right now, as the comet moves toward the sun, the body of the comet is followed by the tail.
There are always two tails, but as the comet nears the sun these diverge, and it is this divergence that enables them to be seen as separate objects.
Actually, if you look closely, I think you can see a third tail.