Electrons (or quasiparticles) pass through the barrier by the process of quantum tunnelling.
Classically, the electron has zero probability of passing through the barrier.
They function as an ohmic electrical contact in the middle of a semiconductor device.
Cooper pairs carry the supercurrent through the barrier by quantum tunneling, a phenomenon known as the Josephson effect.
This setup can form the basis for extremely sensitive magnetometers, known as SQUIDs, as well as many other devices.