The graphite in your pencil is made of carbon atoms arranged in a repeating hexagon pattern called a lattice. The layers of this lattice are very loosely bound, which is why it makes such a great writing implement: The layers shed off as you drag the pencil across paper.
Graphene is what you get if you remove a single layer of graphite, producing a purely two-dimensional material.
CNN has a great interview with the physicists who discovered graphene, along with a great series of infographics that describe some of the amazing properties of this this wonder material and explain what it's useful for.
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