These days the odds of a nuclear war extensive enough to cause our extinction don’t seem as high as in the days when the U.S. and U.S.S.R. faced each other with tens of thousands of large warheads. Even then, although it was impossible to know with confidence what would happen in a full-scale war, it seemed quite possible that bands of hunter-gatherers or small-scale farmers might survive somewhere. Individually or as organized nations, however, we still might be destroyed by nuclear war, and this is certainly a concern for residents of India and Pakistan.
As for other nuclear activities (power plants), they can have very serious accidents, as at Chernobyl, but it is hard to see how that would lead to extinction. Given the unsatisfactory alternatives for energy sources (largely fossil fuels) and the difficulty in getting people to limit either per capita consumption or net population, we may be stuck with more nuclear energy in the future. It will be important to engineer plants with built-in safety, unlike Chernobyl.
Of course nuclear weapons have been around for only sixty years, and while the standoff of the cold war is over, there’s no reason to believe that such a standoff won’t happen again (perhaps with other players). Given the current geopolitical situation, it seems as if small conflicts and incidences of terrorism are much more likely than the big, all-out conflict everyone was afraid of during the cold war, so extinction doesn’t sound plausible in the short run (where "short run" is the next few decades).
Of course, on geological timescales, we’ve only just begun living with these weapons, and we need to all get along at least well enough not to use them. No one can predict the next thousand years, though, and a thousand years is a very short time on the scale of species coming and going (of course with the way we’re going, quite a few species will go extinct in the next thousand years due to non-nuclear human activity). I agree with Mike that the power generation activities do not pose threats like weapons, and that our current consumption of fossil fuels is very risky for a large number of different reasons, including radiation. In fact, a coal-fired power plant releases more radioactive nuclei into the environment than a safe nuclear reactor, just because coal contains uranium and thorium contaminants.
(published on 10/22/2007)