The same phenomenon at the atomic level is called as crossing / moving across Potential Barriers / Wells. These Potential Barriers / Wells are regions of space where there is a sudden increase (Barrier) or decrease in the Potential or there is a presence of an Electric Field in the path of the object.  When a particle like say an electron is moving with an energy E and it encounters a Barrier in its path, a barrier of height V,

Consider the above picture,

    The electron out of sheer curiosity tries to cross the potential and gets trapped inside the potential (It somehow manages to get into the barrier like the boulder climbs the hill). Classically, if the energy E of the electron is lesser than V, then the electron is trapped forever inside the potential and if E > V the electron escapes from the potential. However Quantum Mechanically speaking, the electron escapes the potential even if E < V but if there is a very small difference between E and V.

To understand the Quantum Mechanical picture, let us consider a beach where there are dykes of height H placed along the shore. A cyclone is now raging and the waves have risen up to a height of h. When the waves strike the dykes, then if h < H then the waves do not reach shore. If h > H , then, a portion of the wave reaches the shore, even sometimes the whole wave reaches the shore. We are now interested in those waves that have h < H . Sometimes, these waves will travel at great speeds and though they have h < H they manage to cross the dykes. Thus we see that there is a slight possibility for the waves to cross the dykes even if h < H. Quantum Mechanically speaking there is a sample space that defines the possibility of the waves crossing the dykes. Or in other words, there is a probability though small (but can be considered in Quantum Mechanics) that the wave crosses the dyke. This is the case with the electron crossing the Potential Barrier. Quantum Mechanics says  that every object in this world is a particle as well as a wave. We now consider the electron to be a wave. The amplitude of the wave represents the energy of the particle (E). Even if E < V then Quantum Mechanics says that there is a remote yet considerable chance of finding the particle on the other side  of the barrier. The process used to explain this crossing of the barrier phenomenon is called Tunneling. Tunneling is a process by which the particle wave tunnels through (or makes it way through) the potential barrier to the other side and the probability term that describes this event is called the Tunneling Coefficient.