Only thing I know about superconductors is that here the electrical current face zero resistance. My first question is what is 'super' (physical or mathematical entity) about a superconductor. Or more precisely if I start from the Hamiltonian for a system, which parameter I should try find out to see whether the system is superconducting or not? I read about the mechanism involving the formation of Cooper pair. I understand how they are formed, but don't understand why the pair formation is required for superconductors. Or in other words, how does the pair formation lead to a dissipationless current?
Historically, european were saying supra-conductors, not super-conductor (it is still supraleitung in german, supraconductivité in french, and something similar in russian, but I don't know cyrillic, have fun checking this issue using Wikipedia and changing the language). Only americans were saying super-conductors at the begining (I guess, since I do not really know why the name changed along time, could be interesting to ask on this website if someone knows...). Supra means beyond or something like that : english speakers still use the preffix supra in supra-natural for instance. One usually opposes supra-/super-conductors to the normal ones. So supra-/super-conductivity means the conductivity of the material is not the natural/normal one: supra-/super-conductors do not follow the Ohm's law $j=\sigma E$ relating the current $j$ to the electric field $E$ through the conductivity tensor $\sigma$, as we expect in normal materials. Historically, normal was refering to everything known at that time.
The conceptual reason behind the distinction between super- and normal- conductors, is that a perfect metal should not in principle follow the Ohm's law neither. A perfect/theoretical metal can conduct current perfectly, without resistance. But what was discovered was not only the absence of resistance of a metallic slab at low temperature, but the possibility for persistent current, even large ones. See the website of the Lorentz Institute for some historical perspectives about the discovery by Kammerling Onnes of the disappearance of the resistance and the persistent current. As already suggested in comments, people started to understand the super-conductivity phenomenology only when the Meissner-Ochsenfeld effect was discovered. Superconductors are not only conductors without resistance (as any perfect metal would be as well), they are perfect diamagnetic (they repulse the magentic field from the inside, whereas the perfect metal allow any magnetic field to penetrate the material).
The difference between a normal metal (one should say normal-conductor if we were smart enough to simplify our life) and a superconductor is indeed the charge carrier (I'll come back latter on the reason why I put indeed in italics). In normal metals, the electric current is identified as current of electrons, in superconductors it is a current of Cooper pairs. The Cooper pairs are formed at low temperatures. Nevertheless, one may argue that Cooper pairs are not intrinsically the origin of the superconductivity. Some people prefer to say that the Cooper instability is more important: the fact that the Fermi surface is unstable at low temperature, resulting in a gap at the Fermi energy. This is more or less a semantic problem, since the Cooper instability goes along with the creation of the Cooper pairs as far as we know today. So somehow the Cooper pairs are not required for superconductivity, but come with the Cooper instability. You can read this CERN courier by S. Weinberg about this topic for instance.
Now the subbtle point explaining why the charge carriers are indeed different in normal- and super-conductors. There is in fact no charge carrier in both... more details in a paper by Kivelson and Rohksar from 1990 - PRB 41:11693. But this is far beyond your question I guess.