In short, simple terms a wormhole is a shortcut from one part of the universe to another. You go in one side and come out the other.
If only they were that simple. This post would've been a lot easier.
A wormhole is a point in spacetime where gravity acts as a sort of tunnel, connecting two points in spacetime, so that the trip from point to point through the wormhole is much quicker than without it. To put it into more understandable terms, I like to use John Wheeler's (mentor to Kip Thorne, author of The Science of Interstellar) example of an ant eating though an apple. In this example, the ant's universe is the 2D surface of the apple, and the wormhole is a hole that leads from one side of the surface to the other. The ant could walk around the entire surface, or just straight through the hole. This is the basic idea behind a wormhole: the shortcut through the universe. The apple's insides is not like the rest of the ant's universe, it is in 3D, a dimension above what the ant perceives. For wormholes in our 3D universe, wormholes are 4D, which is exactly why we have to explain them in terms of 2D and 3D.
The first person to answer Einstein's equations that described a wormhole (but Einstein didn't say it was that) was Ludwig Flamm, who described wormholes as spherical shapes that contained no gravitating matter. Much later, John Wheeler and Robert Fuller found that wormholes are created, contracted, and then destroyed. Because of two singularities reaching each other through space and time, a wormhole is created, it expands, shrinks, and then cuts off. They said that this process happens so quickly that not even light has time to make it through before the cut off, and thus nothing can traverse a wormhole.
The only way for travel through a wormhole to be possible is if the wormhole did not cut itself off so quickly, but then there is the issue of bending light rays outward. While something like a sun or black hole can bend rays inward, to bend rays outward would require negative energy. Material with negative energy is called "exotic material" and has actually been created, granted in small amounts, in laboratories (says Thorne).
In short: wormholes are like holes in apples except scaled into a dimension we cannot perceive, and if one were to be traversable we would need a lot of "exotic material" and good fortune.