Quantum tunneling (or just tunneling, or barrier penetration) is a quantum-mechanical effect demonstrating some strangeness of the Heisenberg uncertainty principle, which states that there is a minimum possible product of the certainty of a particle's momentum and the certainty of that particle's position. If a particle is sufficiently close to a sufficiently thin wall (barrier), this uncertainty says there's a chance the particle could be on the other side. A particle approaching one side of the wall could ultimately be found to be on the other side, a phenomenon termed tunneling or barrier penetration. Since there is a finite chance that this happens, with sufficient repeated attempts, the odds of a particle getting through rises to virtual inevitability.
The Heisenberg uncertainty principle is not merely "not knowing where the particle is", but that a "particle's exact location" doesn't always exist. As weird as the effect seems, it is ubiquitous, being an integral part of fusion (thus the shining of the Sun), of radioactivity, and of electronic components including the tunnel diode, which clearly do their function in common consumer devices.
In astronomy, models of stellar structure depend upon a calculation of the fusion rate in a specific plasma according to its temperature and pressure, calculations that include the effect of this tunneling, and stars basically wouldn't shine very much without it.