Consider a small metal ball bearing put in a bowl. The ball bearing has an equilibrium position at the bottom of the bowl. Now if you were to push it a bit it would climb up the walls of the bowl, and fall back again, oscillate about the bottom and come to rest. If you were to push it hard enough however the ball would get out of the bowl.
This is described by saying that the wall of the bowl acts as a potential barrier. The ball is in a potential well. For it to get out you must give it enough kinetic energy (push it hard enough) to get out.
However for very small objects things are not so simple. If the ball had been an electron and the bowl had been a quantum bowl then the ball could have got out without having enough energy to cross the potential barrier. So it is possible for the ball to simply materialize on the other side of the wall (even when it does not have enough energy to cross it) without the wall breaking or rupturing.
This is a very naive explanation of course but I hope it explains the principle behind Quantum Mechanical tunneling.