If two electrons are initially vibrating in unison (a state called coherence) they can remain in wavelike synchronisation even if they are separated by a large distance.
The spins of each electron can be pointed up or down. Lets say that the total spin of the system is zero, so that if the spin of one electron is up, then you know automatically that the spin of the other electron is down.
According to the quantum theory, before you make a measurement, the electron is spinning neither up nor down but exists in a nether state where it is spinning both up and down simultaneously.
Once you make an observation, the wave function "collapses", leaving a particle in a definite state.
1) Physics of the Impossible, Michio Kaku