The International Astronomical Union had met and voted, and the correct answer to that question is now... Eight.
After what some have described as a hotly debated issue, a definition of a planet was produced that included three criteria. It had to be large enough to have a gravitational field strong enough to make it a round object, it had to orbit the sun, and finally it had to have "swept" that orbit of debris (or as the IAU put it, "cleared the neighborhood around its orbit"). Pluto failed because it crossed the orbit of Neptune.
Pluto is demoted to "dwarf" planet, which means the other objects, out in the Kuiper Belt and beyond would also be dwarf planets. Ceres, which held planetary status in the 19th Century, was demoted and then considered for promotion, would be a dwarf planet. Charon would remain a moon of Pluto.
Personally, I find this a little disappointing. I had hoped we were going to expand the Solar System (though I guess we have) and increase the number of planets. In writing science fiction when I have found it necessary to create solar systems in the far reaches of space, they always had more than nine planets (or as is now the case, eight).
The only positive in this is that here, in 2006, we actually debated the issue. I suspect, as our knowledge of the Solar System increases, we may find ourselves in this debate once again. In the Discover magazine article, they suggested that one of the objects out beyond Pluto was the size of Mars. That, it seems to me, would qualify it as a planet (if the size is confirmed), though it might not have the "clean" orbit now claimed as the third part of the definition.
So, now we have answer to the question. How many planets in the Solar System? Eight.