A superconductor/superconducting material is a material that does not resist the flow of electric current through it. This means that they have zero electrical resistance.
Superconductors (such as superconducting wire, for example) are normally refrigerated to very low temperatures to achieve superconductivity, they have an electrical resistance if not refrigerated to the right temperature. They are cooled by cryogenic refrigerators. Cryogenic refrigerators cool to temperatures below -150 °C.
Materials in general, but not all, tend to conduct electricity (as well as heat) better as their temperatures decrease. In other words: electrical and thermal conductivity increase as temperature decreases.
Niobium is an example of a material that can achieve an electric resistance of 0 when it is cooled to a temperature of 9.3 K (Kelvin). Therefore, niobium superconducts electricity at a temperature of 9.3 K. There are other materials which can achieve superconductivity at higher temperatures (which is a good thing because colder refrigerators are more expensive) such as the niobium-tin alloy which superconducts at 17.9 K.
What are Superconductors for?
Superconducting wires for example are sometimes used because they have advantages over other materials such as the ability to transport more electric current per kg of wire. In other words, superconducting wire handles more current than regular wire. Wires in general have a limit to how much current they can transport and when they approach that limit, they become hot.
They become hot because they convert some of the electricity that passes through them into heat, and you probably already realized that is a waste of energy because the electricity which was converted into heat did not make it’s way to the device that needed it, and was converted into heat instead. Fortunately, typical copper wires do not waste much electricity.
One application of superconducting wire that I have seen is in electric motors. Electric motors are usually made of coils of copper wire, but other superconducting wires can be used and cooled to the point of superconductivity which can enable the motors to be manufactured at a fraction of their usual size.
What limits the amount of power an electric motor can handle is the temperature of its coils. They become warmer as more electric current is passed through them and there is a temperature at which the coating on the coil breaks down and short circuits the motor. Therefore, when you supply the motor with more than a certain amount of current, it becomes too hot and breaks down.
Refrigerators not only help to keep the coils cool, but they enable them to superconduct so they actually generate less heat in the first place, so a smaller superconducting motor can handle just as much current as a much larger typical motor.
Improving Power Line Efficiency
Superconducting wire setups can reduce the amount of power wasted in the electricity grid wires. http://en.wikipedia.org/wiki/Electric_power_transmission#Superconducting_cables
So, I hope you understood that two purposes of superconductors is to reduce the size of many devices as well as improve efficiency.