(Disclaimer: This Will Not Happen.)
Aside from widespread panic and confusion, not much. Earth would cool as it does after sunset, and we would be kept warm by the heat retained in the atmosphere, oceans, and land as we are every night.
This is with the assumption that the Sun is simply "frozen" for an hour -- say, a giant bushel is put over it. If you actually turned off all fusion in the Sun, it would collapse and then explode, and then we'd have other fish to fry (although it would take longer than an hour for the explosion to happen). Don't get confused -- the Sun doesn't have enough mass to become a supernova, this would be a different process, which won't happen because you can't turn off all fusion in the Sun. Though there is a small but finite probability that it can stop right now on its own. But it won't. Very probably. Very. But it could.
If the Sun failed to turn back on in an hour (going back to the bushel case now), we would have serious problems. Certainly within a week, the temperature on Earth would have dropped below freezing. People on the coasts might survive longer than the rest, because of the heat the oceans would release; on the other hand I could imagine some intense weather along the coasts due to the temperature gradients. People with large energy reserves would also last longer.
I suppose the place to be would be at the bottom of the ocean, near a geothermal vent (not recommended if you breathe air), where many species may survive for quite some time, perhaps indefinitely?! Some microscopic, one-celled bacteria thrive in the dark, hot and toxic environment of these vents. With no sunlight, these 'archaea' bacteria convert sulphur and other chemicals coming from these vents into energy, much like plants use the sun for photosynthesis. So if the sun didn't turn back on, life on the surface may not like it very much, but bacteria and other extremophiles may not even notice the difference!
This page was last updated on June 27, 2015.