Mining the solar system to build a new world

I watched Armageddon again fairly recently with Bruce Willis, oil drillers in space and an asteroid the size of Texas bearing down on Earth. Buried beneath the Hollywood chaos is a genuinely interesting question: What exactly could we do with an asteroid if we got our hands on one? As it turns out, the answer has nothing to do with blowing it up, sorry Bruce, but everything to do with building a new world.