How do Cooper pairs form? How can Cooper pairs formation be possible, even though it is made up of two likely charged particles? (those particles being electrons)
 A: This question has been asked and answered many times on this site. Below is my own brief attempt at an answer:
The interacton is not strong enough to form bound states. The BCS state is a quantum mechanical state that involves correlated electron pairs (which differs in symmetry from the free electron gas, and leads to spuerconductivity and the Meissner effect). The pairs are strongly correlated in momentum space (involving momenta $\pm \vec{k}_F$, where $|\vec{k}_F|$ is the Fermi momentum), but very extended in coordinate space.
The BCS state does indeed require (weak) attraction, and the main interaction between electrons is the repulsive Coulomb interaction. There are three basic ingredients that conspire to make the effective interaction attractive:
1) The first ingredient is (Debye) screening due to particle-hole pairs: In the electron gas the effective interaction is $V(r)=\alpha/r\cdot\exp(-m_Dr)$, where $m_D\sim e^2k_F^2$ is the Debye mass. At long distances, the Coulomb repulsion is much weaker than in free space.
2) The second ingredient is an attractive phonon (lattice vibration) mediated interaction between electrons. The interaction is attractive for the same reason that all other scalar boson mediated forces are attractive. The coupling constant is small, and depends on details of the system (such as the Fermi momentum and the speed of sound). 
3) The third ingredient is retardation: The Coulomb repulsion is (essentially) instantaneous (the propagator is $1/(k^2+m_D^2)$), but the phonon is retarded (the propagator is $1/(\omega^2-c_s^2k^2)$, because the speed of sound is much smaller than the typical electron velocity. As a result, a suitable pair wave function can avoid the Coulomb repulsion and be sensitive to the phonon attraction. This is sometimes described as the electron being attracted to the "wake" (the lingering lattice distortion) of the other electron.
A: How do Cooper pairs form? 
IMHO it's like Thomas said. As for the electron being attracted to the "wake" of the other electron, IMHO you can get the gist of that via a fluid analogy called the Falaco soliton:
 
The Falaco soliton is a U-tube vorton which is remarkably stable. You make one by dipping a plate into a pool and stroking it forward whilst lifting it clear. It's something like half a Dirac spinor without the bispinor rotation. A better analogy would be a smoke ring which also had a major axis "steering-wheel" rotation on top of the minor-axis rolling rotation. By the by, Thomson and Tait experimented with smoke rings, see this article. They also coined the phrase spherical harmonics. Anyway, you can emulate attraction repulsion and annihilation with Falaco solitons by aiming them at one another at various angles. To emulate Coopers pairs you use two plates to create one Falaco soliton closely followed by another. The second soliton swoops through the inside of the first one and overtakes it. The first one then swoops through the inside of the second, and there's a "dosy doe" combined motion. It doesn't last long, but whilst it does it's reminiscent of dance partners effortlessly swinging each other around. 
