Einstein claimed that "spooky action at a distance" broke relativity by allowing faster-than-light (FTL) communication. If two entangled particles (e.g electrons created as a pair) were separated by a large distance, and a measurement was made on one (like spin), the other particle would simultaneously and instantaneously also have its wavefunction collapse.
Imagine one electron was measured here on earth to be spin-down. This would mean that the other electron, possibly light-years away, would be forced into a spin-up configuration instantaneously. This seems to allow for FTL communication, since there is no delay between the collapses.
The popular resolution is that when we make the measurement on Earth, we cannot choose whether or not the electron is forced into a spin-down or a spin-up configuration; it is random. In addition, there is no way to tell if the other electron has already been measured or not. When we make our 'initial' measurement of spin-down, we don't know if this is due to some alien race already measuring their electron to be spin-up . Therefore, absolutely no information can possibly be transmitted in this fashion.