QM postulates that you can only measure commuting properties at the same time. Position and momentum are non-commuting measurements impossible to measure accurately at the same time.
The attempt of ERP was to use entangled particles to show that you in fact can measure non-commuting states at the same time. That's the paradox.
Take a creation of a entangled electron-positron pair from a gamma-quant. These two particles will take off in opposite directions - same speed. Measuring the position on one particle and momentum on the other, since correlation of the two measurement requires faster that light communication this contradicts relativity. Conclusion is instead that this contradicts QM and that QM is incomplete - missing some hidden variable that would enable us to predict the outcome of measurement.
Real experiments show that this setup doesn't work. The two particles have the same state and measurement on one affect measurement on the other precisely as QM predicts. However a couple of decades was used on developing a hidden variable theory. One of Einstein's less fortunate ideas.
Bells inequality later made a simple and powerful argument to counter any hidden variable theory. But that's another story.