Fullerene is the molecule which is composed entirely of carbon which is also taking the form of a hollow ellipsoid, hollow sphere ring or tube.
Fullerene is very similar in the structure of graphite which is composed of the sheet of linked hexagonal rings but they are also containing the pentagonal rings or sometimes the heptagonal rings that are preventing the sheet from being planar. They are sometimes also called as jocularly buckytubes or buckyballs which depends on the shape. Cylindrical fullerene is often called as the nanotubes. The smallest fullerene is that which do not have any two pentagons sharing the edge and this is destabilizing e.g. pentalene which is C60 and also it is very common.
This molecule was named for Richard Buckminster Fuller who is the noted architect that have created the geodesic dome. Since buckminsterfullerenes are having the similar shape like the dome, the name was thought appropriate.
The structure of C60 is that of a truncated icosahedron which is resembling like a football which is the type made of pentagons and hexagons with a carbon atom at the corners of each hexagon and the bond along each edge. A polymerized single walled nanotubule is the substance which is composed of polymerized fullerenes in which the carbon atoms are forming the one buckytube bond with the carbons in other buckytubes.
Until the twentieth century, diamond and graphite were the only known allotropes of carbon. Then in the molecular beam experiments, discrete peaks were observed which is corresponding to the molecules with the exact mass of 60, 70 or greater numbers of carbon atoms.
Harold Kroto from the University of Sussex, Sean O, Brien, James Health, Richard Smalley and Robert Curl from the Rice University have discovered C60 and the fullerenes. Curl, Kroto, and Smalley were awarded the Nobel Prize of 1996 for their roles in chemistry for the discovery of this class of compounds. C60 and all other fullerenes were later noticed occurring outside the laboratory environment e.g. in normal candle soot. By 1991 it was relatively easy to produce the grams of fullerene powder by making use the technique of Wolfgang Kratschmer and Donald Huffman. In the early 21st century the physical and chemical properties of fullerenes are still under heavy study in both the applied and pure research labs. In April 2003, fullerenes were under the study for the potential medicinal use which is binding the specific antibiotics to the structure to the target resistant bacteria and it is even targeting certain cancer cells such as melanoma.
For more info on the structure of fullerene you can visit MSTnano website where all the details are given with proper diagrams. This information would be helpful in research.