BCS theory explains that in a state of superconductivity, electrons couple to form Cooper pairs. In a superconductor, atoms making up the solid are bonded and do not vibrate independently.  The vibrational energy of molecules in the material behaves similarly to a quantum harmonic oscillator, meaning that energy can only be gained or lost in discrete quantities called phonons. Electrons in the material pair due to the exchange of phonons, which are the quanta of vibrational energy within the crystal lattice. Paired electrons can behave as bosons allowing them to condense into a ground state such that there is an energy gap above the electrons because the wave function of a Cooper pair overlaps with the wave function of other Cooper pairs. The overlap of wavefunctions gives rise to correlated movement between all Cooper pairs in the material; therefore, the superconducting domain will remain in the ground state unless the gap energy is exceeded, and Cooper pairs are broken. One important point to note is that the energy gap in a superconductor is not the same as a band gap, since a superconductor's electrons are condensed into the ground state. Instead, the energy gap describes the energy needed to break up the Cooper pairs. 

Cooper pairs experience attractive interactions.  When one of the electrons moves, it attracts nearby nuclei which are positively charged. The resulting motion temporarily makes a hole in the lattice pulling the second electron in the same direction as the first. The electrons in a Cooper pair must have the opposite momenta and spin of equal magnitudes. Electrons can experience Cooper pairing when they are hundreds of nanometers away from each other. 

The full mechanism of the interaction is a quantum effect governed by phonon-electron interactions. An electron within the lattice can scatter and produce a phonon that can then be absorbed by a second electron allowing for the exchange of momentum between the two electrons. In a superconductor, these interactions are attractive enough to overcome the Coulombic repulsion between the electrons.