The other tail is the dust tail. Like Hansel and Gretel leaving bread crumbs to mark their way through the forest, ISON is leaving a trail of comet dust as it moves through the solar system. Compared to the lightweight molecules in the ion tail, grains of comet dust are heavier and harder for solar wind to push around. The dust tends to stay where it is dropped. The dust tail, therefore, traces the comet's orbit and does not point directly away from the sun as the ion tail does.
Comet ISON is currently moving through the constellation Virgo low in the eastern sky before dawn. Shining like an 8th magnitude star, it is still too dim for naked eye viewing, but an increasingly easy target for backyard optics. Amateur astronomers, if you have a GOTO telescope, enter these coordinates. Special dates of interest are Nov. 17th and 18th when the comet will pass the bright star Spica. Sky maps: Nov. 11, 12, 13, 14, 15, 16, 17, 18, 19.