Graphite, the "lead" in a pencil, is a stack of flat carbon sheets, piled up like the pages of a book. Lift off a single sheet and you have graphene: a sheet of carbon just one atom thick.
In graphene, each carbon atom uses 3 of its four hands to hold 3 neighbours. Three bonds spread out in a flat triangle, 120° apart, and 120° is the corner of a regular hexagon. So the atoms join in hexagons that tile the whole sheet, like a honeycomb.
That leaves each carbon with one outer electron to spare. In benzene's flat ring, the six spare electrons are shared by the whole ring. Graphene does the same thing on a much bigger scale: the spare electrons of all its carbons are shared across the whole sheet. You can think of graphene as benzene rings joined edge to edge in every direction, with no hydrogens in the middle.
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Sticky tape and a Nobel Prize
In 2004, Andre Geim and Konstantin Novoselov at the University of Manchester pressed sticky tape onto a piece of graphite and peeled it off with flakes of graphite on it. Pressing more tape onto the flakes and pulling it away, again and again, split them thinner each time, down to flakes just one sheet thick. On a silicon wafer with a thin glassy coating, even a single sheet shows up under an ordinary light microscope. In 2010 the two of them won the Nobel Prize in Physics for their experiments with graphene.