Superconductors are remarkable materials with a variety of interesting properties, the signature characteristic being nearly zero electrical resistance. The other interesting property is stable levitation (or suspension) of magnetic materials such as iron.
Many electrical components, such as engines, could be made 50-70% lighter and four or more times smaller than conventional engines by using superconductors in their magnetic coils. The problem is finding a room-temperature superconductor -- all known superconductors only display their properties at very low temperatures, below that of liquid nitrogen. If a superconductor could be found that displayed its properties at liquid nitrogen temperature, this would make them practical for use in some parts of industry.
The LHC particle accelerator at CERN uses superconducting coils to accelerate charged particles to velocities within a fraction of a percent of the speed of light.
If room-temperature superconductors could be found and cheaply mass-produced, the cost of electricity might be halved, as at least half of the cost of electricity accounts for the loss accrued by electrical resistance in wires.