A quantum mechanical phenomenon that can be pictured as follows. Imagine a ball rolling towards a hill:
Leaving quantum effects aside (in other words, in classical physics), we expect that what happens depends on the ball’s energy: If the ball moves fast enough (i.e. has sufficient energy), it will climb the hill, pass the peak at B and roll down on the other side. If the ball is too slow, it will reach some maximum height and then begin to roll back down without having passed B.
In the analogous situation for a quantum particle, there is another possibility. Even an incoming particle with enough energy to climb to the height A, but not to pass the peak B, can appear on the right-hand side of the hill at point C and continue onwards. Such a transition is called tunneling – it is as if the particle had taken a secret tunnel from A to C to avoid the forbidden peak around B and arrive directly at C.
More generally, tunneling describes any transition from a state A to a state C that a quantum system can make, but that is forbidden to analogous systems in classical physics, since there, getting from A to C would only be possible by passing through a forbidden state B.