[A planet is] a celestial body that is in orbit around the sun, has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a ... nearly round shape, and has cleared the neighborhood around its orbit.
So we now have 8 planets, MVEMJSUN, a class of "dwarf planets" (which I think are anything round but not unique in orbit), and "small solar system bodies" (anything else).
And to paraphrase my friend Foxtrot, despite what CNN says, this will not CAN not affect the New Horizons mission. It's already been launched.
ETA: Ugh. And despite what else CNN says, Ceres was never a planet. it was temporarily incorrectly called one when we didn't have an official definition. This (BBC) may be a better article.
ETAA: I'm liking this definition less and less. According to CNN, "Pluto is automatically disqualified because its oblong orbit overlaps with Neptune's." Well then, why isn't Neptune disqualified b/c it didn't clear out Pluto, et al.? It could be that Pluto's orbit's different enough that we can let it slide for Neptune's definition and then it's actually Charon and the Plutinos (ooh, a good band name!) that disqualif Pluto. But then what about Jupiter and its Trojan asteroids?