If NASA or another space organization were to shoot an asteroid with a nuclear weapon, what would happen?

It would break up into a bunch of little pieces of radioactive asteroid that would rain down on Earth. That's not a good thing. There's a much better way to deal with this problem - simply push the asteroid out of the way. A small change to it's orbit when it's still far away from Earth will make it miss us entirely. I know Northrup-Grumman was working on a plan for this a few years ago, should it prove necessary.