An Einstein-Rosen bridge, named after the creators of the theory: Albert Einstein and Nathan Rosen, refers to the wormhole inherent in the center of a black hole and is a portal to a mirror universe that exists on the other end of a black hole. But there has been revisions to their theory through the years.
The first revision to this was the Schwarzschild solution, stating that a black hole was a static, non-revolving object, and that the center of a black hole was a single point, meaning that an object caught in that kind of black hole would undoubtedly be crushed when it reached the center, by the infinite gravity contained therein. This revision gives the impression that the Einstein-Rosen bridge would never be a scientific fact.
The second revision came in 1963 when Roy Kerr devised his solution for the Einstein-Rosen bridge equation, which was that if a star was rotating whenever it became a black hole, the center wouldn't become a single point. Instead, it would create a rotating black hole (the kind seen in the film) and there would be a ring instead of a single point at the center. Therefore it could still be possible to traverse through the black hole and emerge through to the other side, under certain circumstances.
The first condition of this is that the object must be travelling faster than the speed of light, in order to prevent being crushed by the finite gravity of the black hole, as detailed by Kerr's solution. The second condition is that the object going through the black hole must have the trajectory of approaching it from the front, head on, instead of the side. Any object approaching the black hole from the side would be crushed by the gravity of the hole itself, which would still be at an infinite state at that point around the black hole. Any object meeting these two conditions would, theoretically, make it through the Einstein-Rosen bridge and reach the universe on the other side of the black hole.
In the film The Black Hole, because the probe ship met those two conditions, it successfully made the voyage through, while the Cygnus, since it didn't meet one or both of the conditions, was crushed and destroyed at the exterior of the black hole itself.