Imagine you have a ball (or any object) that is sitting in a bowl. The bowl has two dips, or wells, that the ball can potentially fall into.
Now, imagine that this ball also has energy - it is vibrating and moving around in the bowl. When the ball is at the bottom of one well, it has low potential energy (energy it has because of its position in the bowl). However, in order for the ball to move up and over to the other well, it needs to overcome a certain amount of energy called a barrier. This barrier is like a hump that the ball needs to climb over, and it requires a lot of energy to get over it.
So, the ball can either stay in the well it's in, or it can potentially jump over to the other well. The double-well potential is the idea that the ball has two potential resting places in the bowl, and it needs enough energy to get over the barrier to move between them.
This double-well potential concept is used in a lot of physics and chemistry applications to explain things like molecular motion and electrical conductivity.