Buckyballs are molecules entirely made of carbon, in the shape of a ball. Constructed carbon molecules in other shapes are called Fullerenes, and can be used to create larger structures.
Buckyballs themselves are not being used to create batteries, but the cylindrical versions of the same carbon molecule, commonly called Carbon Nanotubes, are being used to create "paper batteries," or batteries as thin and flexible as paper. These batteries have the potential of carrying the same charge as a conventional battery in a fraction of the space, resolving both weight and heat issues, and in theory they would be cheaper to replace.
Buckyballs are also being used in solar panels to trap photons and send them along carbon nanotube wires. This could result in a paper battery with a paper solar cell on top, about the thickness of a playing card; this would allow portable devices to easily charge in sunlight without extra bulk.