What would Earth be like without the Moon? If somehow our moon was destroyed (for example by an asteroid) what would happen to Earth? 

Would the Earth be affected by the absence of the moon? Would humans be able to live normal lives without the moon?
Destroying the moon is not easy. It's pretty big. Has quite a lot of gravity and is sitting in the Earth's gravity well. It gets hit by a lot of asteroids and doesn't get destroyed.

The moon has a binding energy of about $10^{29}$ joules. That is a lot of energy. Even if you got the largest asteroid (ceres) to collide with the moon you would not get close to destroying it.

You would need something like the full power of the sun for about 15 minutes. 

If something massive enough to disrupt the moon were to hit it, the energy of that explosion, and the massive amount of moon rocks that would subsequently hit the Earth would probably be enough to melt the surface of the Earth. It would be the end of life on Earth.

There is another way to get rid of the moon. Get a wizard to magic it away. The effect of not having a moon would then be pretty minor in the short term (no tides) and fairly significant in the long term. Human life could continue as normal for a while.

The moon stabilizes the rotational axis of the Earth, without the moon the angle would vary more (over thousands of years) this might mean more extreme ice ages and perhaps ultimately a snowball Earth scenario.