A superconductor is a material with two characteristic properties: zero resistance and the Meissner Effect.
Zero resistance means that it conducts electricity perfectly. When electricity flows through a superconductor, no energy is lost to heat. This is obviously has many practical uses. If we could just replace all of our transmission lines with superconductors, for instance, we could save so much electricity.
What's more, we can create persistent currents in a superconductor. These are currents that just keep on going and going in a loop for an indefinite amount of time. No energy is lost to heat, so the current never stops. One of the practical applications of this are strong magnets. A loop of current creates a magnetic field, and no material can maintain a current as efficiently as superconductors. MRI machines, for instance, use superconducting magnets.
The Meissner Effect is a little trickier to describe.
If you've ever played with a bar magnet, you know that it attracts certain kinds of metals, even if those metals are not themselves magnets. Materials that attract magnets are called paramagnetic. They attract magnets because when they are in the presence of a magnet, they spontaneously form their own magnetic field in the same direction. So when you put a magnet and a metal together, they behave like two magnets attracting each other.
However, many materials are not paramagnetic, but diamagnetic. When they're in the presence of a magnet, they spontaneously form their own magnetic field in the opposite direction. This causes them to be repelled by magnets. Diamagnetism is usually very weak, so it's hard to see just by fooling around with refrigerator magnets.
The Meissner Effect means that a superconductor is a perfect diamagnet. When a superconductor is in the presence of a magnet, it spontaneously forms a magnetic field in the opposite direction, just enough to cancel the magnetic field. So when you put a superconductor near a magnet, the magnetic fields just go around the superconductor, never penetrating through. This also means that superconductors repel magnets, strongly enough to create magnetic levitation.
The lines represent the magnetic field, which is going around the superconducting sphere. (Image courtesy of Geek3)
This is the classic superconductor demonstration of magnetic levitation. (Image courtesy of David Monniaux)
As we now understand it, the reason for superconductivity has to do with fermions and bosons. Fermions are particles that cannot occupy the same state. For example, when you have a lot of electrons in a single atom, each electron must occupy a successively higher energy state, from the core electrons to the outer electrons. Bosons, on the other hand, are allowed to crowd a single state. If electrons were bosons, then they would all collapse to the smallest orbital around an atom.
The reason for fermions and bosons is complicated enough to devote a whole blog post complete with pictures. Let's just say that single electrons are fermions, but paired electrons are bosons. This is what happens in a superconductor. The electrons pair up, and behave like bosons.
Pairing up electrons requires that there be some attractive force between the electrons. This is hard to imagine, because electrons have like charges, and normally repel each other. But remember that we don't just have negatively charged electrons, but positively charged atomic nuclei. If we have an electron, it pulls nearby atomic nuclei towards it, creating an excess positive charge. This positive charge weakly attracts a second electron. The weak attractive force between electrons allows them to pair up.
This is known as BCS theory (named for the physicists who proposed it, Bardeen, Cooper, and Schrieffer). It allows for superconductivity up to about 30 K above absolute zero (that's about -240 Celsius, or -400 Fahrenheit). Above this temperature, the positively charged nuclei fluctuate too much from thermal energy for the mechanism to work properly.
The thing is, since 1986, we have observed superconductors at much higher temperatures! The highest temperature superconductors work up to about 125 K* (-150 Celsius or -240 Fahrenheit), which is still really cold, but too hot for physicists. It's a holy grail for physicists to figure out the correct explanation for high-temperature superconductors. It's a guaranteed Nobel Prize. With a good explanation, we may be able to create superconducting materials at even higher temperatures, perhaps even at room temperature. Once we get room temperature superconductors, we will finally be able to create those perfect transmission lines that we were dreaming of, among other things. Until then, all our superconductors have to be refrigerated by something like liquid nitrogen. Unfortunately, this makes many possible applications of superconductors impractical.
*You might see claims of superconductivity at higher temperatures, but these aren't generally accepted.
One of the interesting things about these high temperature superconductors is that the Meissner Effect does not strictly hold. Superconductors are divided into two types: Type-I and Type-II. Above a certain magnetic field, Type-I superconductors stop behaving like superconductors. This is because the Meissner Effect requires a certain amount of energy. The stronger the magnetic field, the more energy it requires. If the magnetic field is too strong, then it takes too much energy for the material to remain a superconductor. Instead, it just becomes a regular material with electrical resistance.
But all high-temperature superconductors are Type-II. Type-II superconductors remain superconductors even in stronger magnetic fields. Once the magnetic field is strong enough, it starts to penetrate the superconductor in thin tubes.
Magnetic field penetrates Type-II superconductors in thin tubes. (Image courtesy of Hyperphysics)
What's going on here is that the material is now a mix of superconducting and non-superconducting material. The bulk of the material is superconducting, but the thin tubes are non-superconducting. That's why the magnetic field can penetrate through the tubes, but not through the rest of the material.
The difference between a Type-I and Type-II superconductor is that a Type-I superconductor doesn't like having lots of boundaries between superconducting and non-superconducting material. Having lots of little non-superconducting tubes wouldn't make sense in a Type-I superconductor because the boundaries would just require too much energy. On the other hand, in a Type-II superconductor, boundaries don't require extra energy.
As far as practical applications go, Type-II superconductors are probably more useful, since they allow for the strong magnetic fields needed for MRIs, maglev trains, and any large currents.
And right about now, we're at the limit of what can be explained to a popular audience, also incidentally about the limit of my knowledge. But if you have any questions, feel free to ask.