Carbon nanotubes and graphene are two sides of the same coin, in which a specific bonding pattern in carbon lends it semiconducting and conducting properties. Where graphene tends to be thin, flat, large(ish) sheets of conductive carbon, the nanotubes can be thought of like cylinders made by rolling up those sheets. They're pretty cool materials, with all sorts of promise for cheap, flexible, transparent electronic applications. Carbon, after all, is much cheaper than a lot of the noble metals that get used as electrodes.
Folks have been heralding the Coming of Nanotubes for years now. It's one of those techs that perpetually seems 10 years away. Carbon nanotubes have been imagined in all sorts of applications (hell, the topic even has its own wiki), from continued electronic miniaturization, to medical diagnostics and delivery systems, to mechanical reinforcements. At the just-crazy-enough-to-maybe-work end of the spectrum, carbon nanotubes have been proposed as a material potentially strong enough for an eventual space elevator. And the discovery graphene, of course, won the Physics Nobel a few years back, which I was a bit skeptical about.
One of the few things packed with carbon nanotubes that you can actually go out to a store, right now, and buy. Photo from BNC and cozybeehive.blogspot.com.