From Andrew Zimmerman Jones,
Your Guide to Physics.
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Definition: Graphene is a material discovered by a team out of the University of Manchester in 2004. It is a sheet of carbon atoms bound together with double electron bonds (called a sp² bond) in a thin film only one atom thick. Atoms in graphene are arranged in a honeycomb-style lattice pattern.

Structure of Graphene

Graphene is extracted from graphite, which is how it gets its name. In fact, graphene provides the structural basis of all other graphitic materials, from graphite itself to fullerenes (carbon nanotubes, buckeyballs, etc.). Perfect graphene is in hexagonal form, although imperfections can cause heptagonal or pentagonal structures.

Obtaining pure graphene in planar form is difficult and, until 2004, it was assumed by many to be impossible. According to a March 2008 Scientific American, a sample smaller than the width of a human hair would be approximately $1,000, making it currently one of the most expensive materials on the face of the planet.

Graphene in Nanotechnology

Since its discovery, graphene has grown central to much of the research into nanotechnology, due to the unusual electrical, magnetic, and other properties that it possess. It remains to be seen what ultimate use will be obtained from graphene, but research in this area is in extreme demand.