Quantum tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. The classical analog is to go through a wall, which happens in the quantum world but not with typical matter.
One of the major applications is in electron-tunneling microscopes (see scanning tunneling microscope) used to see objects that are too small to see using conventional microscopes. Electron tunneling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunneling electrons.
Tunneling is a source of major current leakage in Very-large-scale integration (VLSI) electronics. This results in the substantial power drain and heating effects that plague high-speed and mobile technology.