What are they? Asteroids are rocky. They come from the inner solar system, where any ice would have been baked off by the sun long ago. Their orbits are fairly predictable. With enough observations, we can track down the big ones and determine if they're threats.
When did an asteroid impact for the last time? The Earth has been getting hit by asteroids and comets for its whole life. The planets formed from collisions of smaller objects, and even our water may have come largely from comets. Heavy bombardments may have continued until as recently as 4 billion years ago,
WHY IS IT SO DESTRUCTIVE? Because while the odds of an impact are small, the number of people who would be killed is so high that the risk (impact odds times the number killed) isn't negligible.
How can we prevent from an impact? The Spaceguard Survey is an attempt to locate and track as many near-Earth objects (NEOs) as possible. Each time astronomers identify an asteroid that isn't on a collision course with Earth, the calculated odds of an impact go down a little bit. By the time the current survey is completed the estimated odds of dying in an asteroid impact will have decreased by a factor of 10, from 1 in 70,000 to 1 in 700,000. Assuming, of course, that they don't find one that really is going to hit.
WHERE IS THE CLOSEST ONE FROM EARTH SITUATED? The closest asteroid from Earth is called "Apophis", Apophis is an asteroid about 270 meters across (almost three football fields). In 2029 it will pass very close to the Earth: within the orbits of our communication satellites. It won't hit; however there is a slight chance that this close pass will shift its orbit exactly the right amount to cause it to hit Earth on a second pass in 2036..