A ball rolling toward a slope with less energy than the height of the slope stops partway and comes back. It cannot go higher than the energy it holds.
The same shape of situation exists for an electron. A stretch where the potential energy is raised is called a barrier. Write its height as U and the energy of the electron as E. If E < U, classical mechanics has no way to the other side.
Getting across would need the kinetic energy inside the barrier to be negative. Kinetic energy is \dfrac{p^2}{2m} and cannot be negative. That is why it is never.
An electron is a wave. The wave function does not fall to 0 in front of the barrier, but enters it.
Inside, however, it does not oscillate. Where E < U, the wave function does not take an oscillating form but one that shrinks the further in it goes. \psi falls off in the form e^{-\kappa x}.
The rate of the falling off is \kappa = \dfrac{2\pi\sqrt{2m(U-E)}}{h}. The higher the barrier, and the heavier the particle, the faster it falls.
If the barrier has a finite thickness, the wave reaches the far edge before it has decayed away. If \psi is not 0 where it arrives, an oscillating wave continues on beyond it.
A wave on the far side means the electron is sometimes found there. It has gone through a barrier it should not have been able to cross. This is called tunneling.
The energy has not changed. Nothing was borrowed inside the barrier to make up a shortfall. It is only that the wave reached the other side.
The fraction that comes out the far side is called the transmission. Since \psi falls off as e^{-\kappa x} and the probability is its square, the transmission T comes to about e^{-2\kappa d}, where d is the thickness of the barrier.
Being an exponential, it drops sharply for a small increase in thickness. Double the thickness and the transmission is not halved but becomes the square of its original value.
This is why it does not happen at everyday sizes. Walls are not walked through, not because it is forbidden, but because \kappa d is larger by orders of magnitude.
Alpha decay is this. The \alpha particle inside a nucleus is trapped by the barrier of the nuclear force. Classically it cannot get out, but it escapes little by little by tunneling. Half-lives differ by orders of magnitude from nucleus to nucleus because the transmission depends exponentially on the thickness.
The scanning tunneling microscope is this as well. The gap between the tip and the sample is the barrier, and the current passing through it is measured. A change of 0.1 nm in the gap changes the current by orders of magnitude, so the bumps of single atoms become visible.
Both of them use the very property that the transmission depends exponentially on the thickness. Because the transmission drops so sharply as the thickness grows, the microscope can measure fine bumps.