BCS theory is the most commonly accepted explanation for superconductivity. The theory, which has been able to describe the behavior of Type I superconductors to a high degree of accuracy, was developed by John Bardeen, Leon Cooper, and Robert Schrieffer in 1957; these three scientists were rewarded for their work 1972 when they were granted the Nobel prize.
BCS theory came into existence as a result of several key observations that were realized in the mid-1900s, and which necessitated a rethinking of the general phenomenon of superconductivity in terms of quantum physics. Specifically, the critical temperature required for the superconducting-normal phase transition clued scientists in to the idea that some sort of an energy band gap was responsible for the transition, and pointed to something similar to Bose-Einstein condensation. (This brought about another problem, however, in that electrons are ½-spin particles that must behave as Fermions, precluding the Bose-Einstein explanation.) The other key observation was one which linked the critical temperature for the transition with the nature of the material; specifically, scientists found that different isotopes of the same material exhibited slightly different critical temperatures, highlighting the importance of the structure of the lattice in the conductor to its ability to behave as a superconductor.
The product of these two observations was, essentially, BCS theory. Bardeen, Cooper, and Schrieffer postulated that as an electron moved through the material of the conductor, it dragged along with it a string of positively charged nuclei due to Coulomb attraction. While this interaction was not strong enough to take the nuclei very far, it did displace them slightly from their assigned places in the lattice. These nuclei in turn would attract an additional electron, to be pulled along in the first electron's wake. When coupled together, the two electrons formed what would come to be referred to as a Cooper pair; the pair would collectively have an integral spin number, allowing all of the pairs in the material to behave as Bosons and to condense into a single state . As such, when the thermal energy that kept the material in the normal phase dropped below the band gap (ie, the temperature fell below Tc), the material would almost instantaneously go superconducting.
BCS theory has been confirmed on a number of fronts. The isotope effect proves that heavier isotopes of a given element require subsequently lower and lower critical temperatures in order to attain the superconducting state; this indicates that, since a heavier nucleus will necessitate a larger disturbing force to displace it from the lattice, additional energy is required to make the superconducting jump. Further, good conductors such as gold and silver do not appear to a have a superconducting state at all, which gives great credence to BCS. While it would make sense for better conductors to make even better superconductors, BCS theory states that it is the phonon interaction (that taking place between the lattice and the metal's electrons) which is key to superconductivity, and it is the very weakness of this interaction that makes good conductors so good -- and, we now find, which keeps them from becoming 'super'. :P