|MadSci Network: Astronomy|
If an object as big as the Moon hit Earth, this would most certainly be an Extinction Level Event (ELE). The amount of damage depends on how big the object is and how fast it is moving. In general, partial extinctions, similar to what happened to the dinosaurs, would occur with an object about 5 km (3 miles) across. Surface sterilization (where surface life goes extinct but deep ocean life survives) would happen with an asteroid about 50 km (30 miles) across. Total destruction of the environment, vaporization of the oceans, and sterilization of the planet would occur with an asteroid larger than about 100 km (60 miles). By contrast, the Moon is almost 3500 km (2200 miles) across!
It's very hard to change the motion of an object as big as the Moon. To change to Moon's orbit, it would have to move faster or slower. It takes a lot of energy to do this. What if we smashed an asteroid into the Moon? A lot of asteroids have hit the Moon in the past (just look at all the craters). Most asteroids are very small compared to the size of the Moon and aren't moving fast enough to make the kind of changes that would cause us to have to worry about the Moon hitting us. Usually they break apart and break pieces of the Moon off, too. Most of these pieces fall back on the Moon forming ejecta rays (light colored regions spreading out from a crater which can be seen on a picture of the Moon). A few might escape the Moon's gravity and eventually fall to Earth. Whether or not they cause any damage depends on how big they are.
You need not worry about the Moon hitting Earth. It's more likely we'd get hit by an asteroid or other debris in space. There are always small dust particles and tiny pebbles or rocks falling on Earth, but most of them are vaporized in the atmosphere (meteors) or fall harmlessly to the surface (meteorites). Bigger objects are much less common. The last planet sterilizing event happened early in the solar system, probably not long after the Moon was formed. Extinction events are very rare, and the chance of one occurring in our lifetime is very small.
There are groups of astronomers around the world who are working on finding and cataloging all the objects in the solar system which cross Earth's orbit and could potentially cause damage. If you'd like to learn more about the search for NEOs (Near Earth Objects) and impact hazards, check out Near Earth Impact Hazards at NASA's Spacelink and the Near Earth Object Program at JPL.
Try the links in the MadSci Library for more information on Astronomy.