Wormholes on Kruskal-Szekeres coordinates 
Kruskal-Szekeres coordinates in the diagram are
$$v=\Big(\frac{r}{2M}-1\Big)^{1/2}e^{r/4M}\sinh\Big(\frac{t}{4M}\Big)$$
$$u=\Big(\frac{r}{2M}-1\Big)^{1/2}e^{r/4M}\cosh\Big(\frac{t}{4M}\Big)$$
Region-$\mathrm{I}$ is the interior of our universe and exterior of a black hole interior in region-$\mathrm{II}$, region-$\mathrm{III}$ is the interior of another universe and region-$\mathrm{IV}$ is the interior of a white hole. Those $45^{\circ}$ lines are the event horizon and darker hyperbolas in region-$\mathrm{II}$ and region-$\mathrm{IV}$ are the singularities.
Wormholes are said to be tunnels connecting two different parts of the same universe or connecting two parts in different universes.
Einstein-Rosen bridge connects the region-$\mathrm{I}$ and region-$\mathrm{III}$ so particle passes through the horizon and appears on the other side of the universe, this makes sense but I don't see how Einstein-Rosen bridge can connect two different points in the same universe.
 A: Here are two possible embedding diagrams for a wormhole:

In the first one, the wormhole connects two different universes. In the second one, we simply glue together the two sheets. This can be done as a purely topological maneuver; although the connection looks curved in the embedding diagram, it has no intrinsic curvature.
Misner, Thorne, and Wheeler have a discussion of this kind of thing on p. 837.
All the material in the answer by SuperCiocia is wrong. The first part is wrong because it ignores the topological issue described above. The second part is this:

It is also worth to point out that Schwarzschild metric solution, which this diagram and reasoning is based on, is a solution for the exterior of the black hole. For the interior of the black hole, which one would have to cross in order to pass through this bridge, the solution might well be different and not lead to such a bridge in the first place.

This is also wrong. The question clearly asks about the Schwarzschild spacetime in Kruskal-Szekeres coordinates, the whole point of which is to get the maximal extension of the spacetime. In general, it's a bad idea to be in a hurry to accept an answer on SE.
A: The two spacetime regions $u<0$ and $u>0$ are always spacelike separated and hence can never have any contact. Hence why they are referred to as "different universes", because in the same universe you would have at least some causal contact between regions at some point.
It is also worth to point out that Schwarzschild metric solution, which this diagram and reasoning is based on, is a solution for the exterior of the black hole. For the interior of the black hole, which one would have to cross in order to pass through this bridge, the solution might well be different and not lead to such a bridge in the first place.
