Why do superconductors conduct electricity without resistance?

A superconductor conducts electricity without resistance because the supercurrent is a collective motion of all the Cooper pairs present.

In a regular metal the electrons more or less move independenly. Each electron carries a current $-e \textbf{v}(\textbf{k})$, where $\textbf{k}$ is its momentum and $\textbf{v}(\textbf{k}) = \partial E(\textbf{k})/\partial \textbf{k}$ is the semiclassical velocity. If an electron gets scattered from momentum $\textbf{k}$ to $\textbf{k}'$ it gives a corresponding change in the current. A sequence of such processes can cause the current to degrade.

In a superconductor, the story is totally different because the Cooper pairs are bosons and are condensed. This means that Cooper pairs self-organize into a non-trivial collective state, which can be characterized by an order parameter $\langle \Psi(\textbf{x}) \rangle = \sqrt{n} e^{i\theta(\textbf{x})}$ (where $\Psi$ is the annihilation operator for Cooper pairs.) which varies smoothly in space. Since the current operator can be written in terms of $\Psi$ it follows that gradients of $\theta$ give rise to currents of the condensate: $\textbf{j} = n(\nabla \theta + \textbf{A})$. All the small-scale physics (such as scattering) gets absorbed into the effective macroscopic dynamics of this order parameter (Landau-Ginzburg theory).

One should think of every single Cooper pair in the system taking part in some kind of delicate quantum dance, with the net effect being a current flow. But this dance is a collective effect and so it's not sensitive to adding or removing a few Cooper pairs. Therefore, scattering processes don't affect the current.

A superconductor is characterized by two main properties:

  1. zero resistivity, and
  2. the Meissner effect.

Equivalently, these can be stated more succinctly as

  1. $E = 0$ (remember that resistivity is defined as $\frac{E}{j}$), and
  2. $B = 0$.

So even more succinctly: superconductors are characterized by no internal electromagnetic fields!

What is the intuitive reason for this? It can be understood from the fundamental/microscopic property of superconductors: superconductors can be described in terms of superpositions of electrons and holes. Note that these two components have different electric charges, hence such a superposition can only be coherent if nothing couples to the charges inside a SC! Indeed, if there were an electromagnetic field inside the SC, it would couple differently to the electron and hole, decohering the superposition and destroying the SC. [Of course this doesn't do full justice to the theory of superconductivity, since this reasoning doesn't explain why we have superpositions of holes and electrons. Rather, my point is that once we start from that, then the aforementioned is hopefully intuitive.]