If the sun were to suddenly be extinguished, the planets in our solar system would become much colder. The earth would almost instantaneously go into an ice age.

Scientists believe that the actual fate of the sun is that it will run out of hydrogen in the core which it uses as fuel to burn. When this happens some more hydrogen at the surface will fuse causing the upper layers of the sun's surface to expand. It will expand so greatly that its radius will overcome the earth vaporizing our planet. After this the sun may begin to cool (lasting a few billion years) as for the fate of other planets farther out than earth, I am unsure.

And of course, there is no reason for you to worry about the end of the sun since the amount of hydrogen left to burn will last for about another 10 billion years.

The sun would have to cool down before anything would change. This would take a long time before we would even notice - thousands of years, and millions of years before the sun would genuinely cool off.

As this took place, the solar system would freeze because of the loss of energy input. Nonetheless, all of the planets, moons, asteroids, comets, etc. in the solar system, would continue moving as they have always done since the sun's gravity is determined by its mass, which is not affected by its being turned off.