The theory behind the substance graphene was first explored by theoretical physicist Philip Wallace in 1947 as kind of a starting point when he was doing research trying to understand the electronic properties of more complex, 3D graphite. although the name graphene wasn't actually coined until 40 years later, where it was used to describe single sheets of graphite. In other words, it's the name given to a flat monolayer of carbon atoms that are tightly packed into a 2D honeycomb lattice; like a molecular chicken-wire that is one atom thick. It's essentially the basic building block for graphitic materials of all other dimensionalities; it's a stepping stone to building bigger things. Graphene in itself however wasn't discovered until 2004 in its full observable and testable form.
Since then, in the past 6 years, scientists have discovered that the substance retains some amazing properties. Some say that it will be heralded as one of the materials that will literally change our lives in the 21st century. Not only is graphene the thinnest possible material that is feasible, but it's also about 200 times stronger than steel and conducts electricity better than any material known to man—at room temperature. Researchers at Columbia University's Fu Foundation School of Engineering who proved that graphene is the strongest material ever measured said that "It would take an elephant, balanced on a pencil, to break through a sheet of graphene the thickness of Saran Wrap."
Potential applications for the material include the replacing of carbon fibers in composite materials to eventually aid in the production of lighter aircraft and satellites; replacing silicon in transistors; embedding the material in plastics to enable them to conduct electricity; graphene-based sensors could sniff out dangerous molecules; increasing the efficiency of electric batteries by use of graphene powder; optoelectronics; stiffer-stronger-lighter plastics; leak-tight, plastic containers that keep food fresh for weeks; transparent conductive coatings for solar cells and displays; stronger wind turbines; stronger medical implants; better sports equipment; supercapacitors; improved conductivity of materials; high-power high frequency electronic devices; artificial membranes for separating two liquid reservoirs; advancements in touchscreens; LCD's; OLED's; graphene nanoribbons could be a way to construct ballistic transistors; and nanogaps in graphene sheets may potentially provide a new technique for rapid DNA sequencing.