Whereas EPR stands for Einstein-Podolsky-Rosen (and one should add Bohm and Bell) quantum correlation between two entangled particles or the so-called Bell pair. So this side of the equation represents quantum mechanics.

These two effects seem both to challenge the locality of spacetime, i.e. our inability to send signals faster than the speed of light. However, it is generally argued that EPR correlations can not be used to send information faster than the speed of light while the ER bridge is generally a not traversable Lorentzian wormehole.

Thus quantum entangled particles are connected by a wormhole. Or more precisely, the ER bridge between two black holes is actually created by a quantum EPR correlations between the microstates of the two black holes. This is a conjecture put forward original by Susskind and Maldacena in 2013. See also Israel in 1976. In this conjecture it is seen that the ER bridge is a special kind of EPR correlations in which quantum entanglement can be described by a weakly interacting gravity theory which is precisely Einstein theory here. Conversely, it is hopped that every EPR correlated state (even the singlet state of two spins) is connected by some ER bridge which is not necessarily the one given by general relativity. Thus the geometry of spacetime and gravity is determined by quantum entanglement.

Some of the early work leading to this conjecture I mention van Raamsdonk in 2010 which states that a maximally extended AdS black hole, which is a non-traversable wormhole, is dual via the celebrated AdS/CFT correspondence to a pair of maximally entangled thermal conformal field theories.

This conjecture is also in some sense a competing proposal for the resolution of the information loss problem (although a much more profound one) to the AMPS firewall.

Recall that a Hawking particle $B$ must be entangled with the early part $R_B$ of the Hawking radiation and with the particle $A$ which went behind the horizon (complementarity). This is however forbidden by monogamy. The AMPS resolution is via a loss of entanglement by breaking the entanglement between $B$ and $A$. Since $A$ is a high energy mode the breaking will release a firewall. However, the mechanism of this breaking is unknown. The interior of a black hole (just inside the event horizon) is thus not smooth.

However, in the ER=EPR conjecture the entanglement between $A$ and $B$ is possible via a wormhole, i.e. an ER bridge. In fact the ER=EPR conjecture states that the Hawking radiation is "connected by very quantum ER bridges to itself and also to the black hole horizon". This can also be seen by collecting all Hawking radiation and then collapsing it into a second black hole. The two black holes are then connected by an ER bridge and the interior of the black hole is seen to be smooth.

References:

1-Einstein, Rosen."The Particle Problem in the General Theory of Relativity".

2-Einstein, Podolsky, Rosen. "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?".

3-Maldacena, Susskind. "Cool horizons for entangled black holes"

4-Israel, Thermo field dynamics of black holes.

5-van Raamsdonk. "Building up spacetime with quantum entanglement".

6-Susskind. "Copenhagen vs Everett, Teleportation, and ER=EPR".

7-Susskind."ER=EPR, GHZ, and the consistency of quantum measurements"