ZapperZ's recent post about new work on the pseudogap in high temperature superconductors has made me think about how to try to explain something like this to scientifically literate nonspecialists. Here's an attempt, starting from almost a high school chemistry angle. Chemists (and spectroscopists) like energy level diagrams. You know - like this one - where a horizontal line at a certain height indicates the existence of a particular (electronic) energy level for a system at some energy. The higher up the line, the higher the energy. In extended solid state systems, there are usually many, many levels. That means that an energy level diagram would have zillions of horizontal lines. These tend to group into bands, regions of energy with many energy levels, separated by gaps, regions of energy with no levels.
Let's take the simplest situation first, where the energies of those levels don't depend on how many electrons we actually have. This is equivalent to turning off the electron-electron interaction. The arrangement of atoms gives us some distribution of levels, and we just start filling it up (from the bottom up, if we care about the lowest energy states of the system; remember, electrons can be spin-up or spin-down, meaning that each (spatial state) level can in principle hold two electrons). There's some highest occupied level, and some lowest unoccupied level. We care about whether the highest occupied level is right up against an energy gap, because that drastically affects many things we can measure. If our filled up system is gapped, that means that the energetically cheapest (electronic) excitation of that system is the gap energy. Having gaps also restricts what processes can happen, since any quantum mechanical process has to take the system from some initial state to some final state. If there's no final state available that satisfies energy conservation, for example, the process can't happen. This means we can map out the gaps in the system by various spectroscopy experiments (e.g., photoemission; tunneling).
So, what happens in systems where the electron-electron interaction does matter a lot? In that case, you should think of the energy levels as rearranging and redistributing themselves depending on how many electrons are in the system. This all has to happen self-consistently. One particularly famous example of what can happen is the Mott insulating state. (Strictly speaking, I'm going to describe a version of this related to the Hubbard model.) Suppose there are N real-space sites, and N electrons to place in there. In the noninteracting case, the highest occupied level would not be near a gap - it would be in the middle of a band. Because the electrons can shuffle around in space without any particular cost to doubly occupying a site, the system would be a metal. However, suppose it costs an energy U to park two electrons on any site. The lowest energy state of the whole system would be each of the N sites occupied by one electron, with an energy gap of U separating that ground state from the first excited state. So, in the presence of strong interactions, at exactly "half-filling", you can end up with a gap. Even without this lattice site picture, in the presence of disorder, it's possible to see signs of the formation of a gap near the highest occupied level (for experts, in the weak disorder limit, this is the Altshuler-Aronov reduction in the density of states; in the strong disorder limit, it's the Efros-Shklovskii Coulomb gap).
Another kind of gap exists in the superconducting state. There is an energy gap between the superconducting ground state and the low lying excitations. In the high temperature superconductors, that gap is a bit weird, since there actually are low-lying excitations that correspond to electrons with very specific amounts of momentum ("nodal quasiparticles").
A pseudogap is more subtle. There isn't a "hard" gap, with zero states in it. Instead, the number of states near the highest occupied level is depressed relative to noninteracting expectations. That reduction and how it varies as a function of energy can tell you a lot about the underlying physics. One complicated aspect of high temperature superconductors is the existence of such a pseudogap well above the superconducting transition temperature. In conventional superconductors (e.g., lead), this doesn't exist. So, the question has been lingering for 25 years now, is the pseudogap the sign of incipient superconductivity (i.e., electrons are already pairing up, but they lack the special coherence required for actual superconductivity), or is it a sign of something else, perhaps something competing with superconductivity? That's still a huge question out there, complicated by the fact that doping the high-Tc materials to be superconductors adds disorder to the problem.