Somebody other than I really needs to do this topic justice. Physicists, lend us your pens! In the interim, however...
A Josephson Junction is a particular type of electronic connection. It is named for the physicist who predicted the behavior of such an arrangement, the Nobel prize winner Brian David Josephson. At its most basic, a Josephson Junction contains two superconductors which are separated by a non-conductive (insulating) barrier. There are two different cases in which the Junction will be useful.
If one side of the barrier (one superconductor) has DC applied to it - direct current - there should, by classical physics, be no way for that current to traverse the boundary of the insulator (at least, not without damaging or destroying the insulator and arcing). There is no conductive path across the gap. However, according to quantum mechanics, it should be possible (and expected) for some portion of the electrons reaching the barrier to tunnel across the barrier. If electrons are viewed as a wave phenomenon, then when the electrons reach the end of one superconductor, there is a finite but definite 'chance' that some of them are 'located' on the other side of the barrier. Viewing them as wave phenomena means that their location is in fact a probability field, and some part of the resulting field will stretch across the barrier. Sure enough, in real-world use, a portion of electrons that correlates to that component of the probability field does, in fact, end up 'across' the barrier, and continues on down the second superconductor.
This is useful not only as a quantum physics experiment, but because arbitrary numbers of electrons can be 'guided' to the second superconductor by solving the field mathematics for the number of electrons and the size of the gap. In fact, single electron transistors can be (and are) built by using Josephson Junctions - devices which switch a single electron across such a gap.
If alternating current is used, a fluctuating magnetic field will be formed around one superconductor, and that magnetic field will induce electrical current in the matching superconductor, since magnetic fields can cross non-conductive barriers. This is useful in practice because the frequency of the fluctuation will always result in a particular voltage across the gap. In essence, the Junction serves as a perfect frequency-to-voltage converter.
BlackPawn notes that this second case is somewhat similar to the use of a capacitor - however, it's not the same. Capacitors are intended to 'store' charge until a stored energy threshold has been reached, at which point the charge has a high enough voltage to cross the built-in barrier and discharge across the circuit. Although the very basic form of the capacitor and the Junction is the same - two conductors separated by an insulator - there are important differences. The Junction uses superconductors, and at no point is a 'stored' charge intended to directly cross the insulation barrier - it is only supposed to traverse the gap via magnetic field induction or a small subset of the charge is intended to tunnel across. The Junction does not rely on the charge building up in the circuit - indeed, it's not supposed to. Capacitors include plates intended to collect electrons and hold them until the differential across the gap becomes great enough to overcome the set resistance of the circuit. (Again, I'm not an EE. Correct me at will.)
This writeup is my own poor understanding of the physics involved. Physicists, please, add content and supersede this.