Quantum tunnelling (or tunneling) is the quantum-mechanical effect of transitioning through a classically-forbidden energy state.
Consider rolling a ball up a hill.
If the ball is not given enough velocity, then it will not roll over the hill.
This makes sense classically.
But in quantum mechanics, objects do not behave like classical objects, such as balls, do.
On a quantum scale, objects exhibit wavelike behavior.
For a quantum particle moving against a potential hill, the wave function describing the particle can extend to the other side of the hill.
This wave represents the probability of finding the particle in a certain location, meaning that the particle has the possibility of being detected on the other side of the hill.
This behavior is called tunneling; it is as if the particle has 'dug' through the potential hill.