Comet ISON is comet that is currently approaching the solar system, and is expected to reach perigee with the sun sometime in late November or early December of 2013. The current date predicted is November 28, 2013. Comet ISON is an Oort Cloud comet that is making its first entry into the inner solar system, from off the plane of the ecliptic. As of the current writing (November 4, 2013), Comet ISON is in the constellation of Leo, and is at a magnitude of 8 or 9, meaning that it is not yet visible to the naked eye, but is easy to see when using a common telescope.
After Comet ISON reaches perigee and swings back around, it should be "incredibly bright"--- if it survives. Alas, comets are mostly made out of ice, and having something made out of ice pass within a million miles of the sun (about 1% of the Earth's distance from the sun) is not a recipe for longevity. But if it survives, it should be very bright. NASA doesn't describe exactly how bright (since comets are unpredictable), but it could easily be the brightest thing in the night time sky, or possibly bright enough to see in the daytime. This would be quite unusual: only nine comets in the recorded history of astronomy have been seen during daytime. Given the rarity and spectacular nature of this comet, it is odd that its arrival has not been trumpeted more in the general media. Perhaps this is because expecting too much of a comet is a good step towards disappointment: consider how Halley's Comet fizzled out in 1986.
However, if the comet does survive its perigee and does maintain its brightness, it will probably come to widespread popular attention in December of 2013. Some of which will probably be of a pseudoscientific variety: I imagine that at least some people will predict disaster from the comet, of either a physical or spiritual nature. But hopefully the rest of us can sit back and watch the show.
: The official NASA site on the comet.
: Sky and Telescope's weekly observing chart, which should have updates on when and how to view the comet.