The EPR paradox (Einstein-Podolsky-Rosen paradox) is a set of experiments that suggest 'spooky action at a distance' because particles appear to share information instantaneously, at a distance, long after an interaction between them.

People applying "common sense" would like to argue that there is some way that the information is being shared -- some hidden variable that collects and shares the information between them.

Bell's Inequality only assumes there there is some such hidden variable operating locally* -- with no specifications of any kind on how it works -- and deduces correlations between particles sharing information that is in contradiction with experiments.

* that is, mechanically rather than 'magically' at a distance