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Go to the art store and buy a good-quality, broad-tipped architect's pencil. Sharpen it to expose a nice, flat expanse of graphite. Then touch it with the sticky side of a piece of adhesive tape. Now pull the tape off: On it will be traces of graphene—a one-atom-thick, barely visible, slightly grayish, marginally iridescent flat network of carbon hexagons.
Diamond, the more-glamorous and higher-strung allotrope of carbon, has a three-dimensional, "face-centered cubic" structure, formed out of linked carbon tetrahedrons (groups of five carbons forming a four-sided pyramid) joined by short, strong covalent bonds (sp3 bonds) all the same length. The result is a so-called "network solid" with awesome physical properties, including incredible hardness—resistance to distortion in all directions—total non-conductivity, and the little-understood but much-admired ability to move relationships from the "going steady" to the "engaged" phase.
Graphene is like diamond when it goes home for the weekend and dials it down a little. Here, instead of tetrahedrons, you have hexagons: rings of six carbon atoms lying in a single plane, joined by slightly longer and weaker sp2 covalent bonds—each carbon waving a free electron around. The free electrons can be delocalized, so graphene conducts electricity in the planar dimension.
In fact, it does so very well (essentially better than anything, and at room temperature, no less). And—bonus!—like its geometrically-kinda-similar brother-from-another-mother, silicon (a.k.a. "the other network solid"), graphene conductivity can be modulated by doping and manipulated by nano-engineering to be very, very spooky. So graphene is now at the center of a vast storm of R&D cashflow around semiconductor, spintronics and quantum-computing applications.
The stuff is also quite strong, courtesy of those sp2 covalent bonds. Defect-free graphene sheets are the strongest material known (and, some chemists argue, the strongest material possible). So experimenters are fooling around, figuring out how to produce defect-free graphene sheets at scale, and how to orient and bond them with plastics and resins to create usable macromaterials.
So it's graphene's time to shine. And honoring the fact, this year's Nobel Prize for physics went to Andre Geim and Konstantin Novoselov of Manchester University (U.K.), who in 2004 exfoliated graphene sheets from ultrapure graphite using the Scotch-tape method described above (who says you can't perform world-class physics experiments at home?), identified it as a heretofore unknown but naturally occurring macroscopically stable 2D allotrope of carbon, and characterized many of its amazing properties.

