The phenomenon in which a superconductor expels external magnetic fields is the Meissner effect.
When a material is cooled below its superconducting transition temperature in a sufficiently weak magnetic field, surface screening currents push nearly all magnetic flux out of its interior. The field is not abruptly zero right at the surface: it penetrates a short distance called the London penetration depth and then falls away approximately exponentially. A superconductor in this state is said to be in the Meissner state.
Walther Meissner and Robert Ochsenfeld discovered the effect in 1933 while studying superconducting tin and lead. Their result showed that superconductors are more than merely perfect conductors. Zero resistance alone would preserve existing magnetic flux, whereas the Meissner effect describes active magnetic-field expulsion during the transition into superconductivity.
The effect is often demonstrated by magnetic levitation, but levitation alone is not definitive evidence: flux pinning and ordinary magnetic forces can also stabilize a magnet. Cooper pairs help explain superconductivity microscopically, while the Josephson effect concerns tunnelling between superconductors. Strong enough magnetic fields can destroy the superconducting state, so “all” means nearly complete expulsion under appropriate conditions.