How can Ohm's law be correct if superconductors have 0 resistivity?
Ohm's law is generally NOT correct, it's called a law for historical reasons only!! It's a law in the same sense in which Hooke's law is a law... it holds only for certain systems under certain conditions, but it's widely known because it's simple and linear!
It's not just superconductors, diodes are a neat everyday example of Ohm's law failing to hold. But it fails for every material under sufficiently extreme circumstances.
Check out this I-V graph for a diode.
Ohm's law works for ordinary conductors for a reason: the particles carrying the current (usually, but not always electrons) scatter incoherently and inelastically from features of the conductor. In the case of an electron current, at low temperature this scattering is caused by impurities in the conductor; at high temperatures, the dominant source of scattering is electron-phonon scattering (phonons are coherent vibrations of the conductors fundamental lattice). As long as those conditions pertain, you can expect Ohm's law to be a good approximation to the behavior of the current.
However, in a superconductor quantum mechanics rears its elegant head and generates a situation in which coherent effect dominate to a point where is effectively no inelastic scattering and as such no energy loss in the flow of current.
Nor are these the only possible situations, as Schlomo Steinbergerstein notes semiconductors exhibit a wide range of conduction behaviors.
The difference in macroscopic physics is down to a difference in microscopic physics.
This business where regimes dominated by coherent and incoherent interaction show very different behavior comes up a lot in various corners of physics and one could spend a long (and possibly enjoyable) time surveying those effects alone.
Looking at your question from the perspective of ideal circuit theory, an ideal resistor has the following I-V relationship:
$V_R = I_R R$
The voltage across the resistor is proportional to the current through the resistor with constant of proportionality equal to $R$.
In ideal circuit theory, an ideal conductor can be thought of as a "zero ohm resistor". Setting $R = 0$ in the above equation gives:
In other words, for any value of current, the voltage across an ideal conductor is exactly $0V$.
Of course, in the real world, there are no ideal resistors or conductors. However, Ohm's Law is still a good approximation for many materials over some limited operating range.
The simplest method to measure the electrical resistance of a sample of some material is to place it in an electrical circuit in series with a current source I and measure the resulting voltage V across the sample. The resistance of the sample is given by Ohm's law as R = V / I. If the voltage is zero, this means that the resistance is zero.