First, let me say that I'm very happy to see NASA taking the route of a prize, much like the X prize, to promote a speculative new technology. For a relatively small sum (at least by U.S. government agency standards!), they will either harness the competitive energies of the civilian sector, or they will get (for free) the collective judgment that the objective is impossible. Good move, and very un-government-like. I applaud it.
In the specific case of tethers, they've set a very tough objective — especially because the primary objective of the prize is a space elevator (a tether from the Earth's surface to a satellite in geosynchronous orbit). If all of this is just blarney and blather to you, take a gander at this short Wikipedia article on the subject.
Personally, my main concern with a space elevator tether system is their vulnerability to micro-meteorite and "space junk". Lives and an investment of billions of dollars could be taken out by an empty Coke can (figuratively speaking) whizzing around in orbit. Not a pretty picture...