A wave packet partially penetrates a barrier which is much higher than the kinetic energy. The wave function decreases exponentially in the "classically forbidden region" but eventually reaches through a thin barrier into a region where it can propagate freely again. This effect cannot be understood with classical physics. A classical particle would always be reflected at a barrier which is higher than the kinetic energy. Quantum mechanically, there is always a nonvanishing probability of tunneling through a finite barrier.
"Visual Quantum Mechanics - Book One" explains an important application of this effect: The scanning tunneling microscope (STM). The STM is an exciting new technology allowing to obtain images of single atoms on the surface of metals.