“Probability amplitude” is a term used in quantum physics. It’s a number that appears in Schrodinger’s Wave Equation. Let’s say, for example, that we want to know the energy level of an electron moving around the nucleus. The electron in a hydrogen atom, for example, might be at the lowest energy level or, alternatively, a little higher energy level. If it’s at the lowest energy level, it would be detectable within a sphere close to the nucleus. At a little bit higher energy level, it might be detected within a larger sphere. There are actually quite a number of energy levels that it could be at, and each level is associated with a different region in which the electron might be detected. (These regions are called “orbitals.”)

To find the energy level, we crank through Schrodinger’s Wave Equation. The probability amplitude which the equation outputs is proportional to the likelihood that, when detected, the electron will be found at a particular energy level. Specifically, if we square the probability amplitude associated with a particular position, it will give us the probability of the electron being detected at that energy level.

Schrodinger’s Equation tells us much more about the electron than its probable energy level. The equation can also be used to calculate its momentum, spin, and several other properties. A different version of the equation is used to calculate each of these properties; each version is described as being in a different “basis”—the energy level basis, momentum basis, and so on.

When Schrodinger’s Equation is in, for example, the momentum basis, the probability amplitude outputted is proportional to the probability of the electron being at a particular momentum.

When we are looking for energy level, the probability amplitude is the square root of the probability that the particle will be detected in a particular position. A weird aspect of the probability amplitude is that it is a complex number. That is, one part of the number is an ordinary (“real”) number, and the other part is “imaginary.” Imaginary numbers are based on the square root of -1 (minus one). The square root of -1 doesn’t have physical meaning in our universe, yet there it is in Schrodinger’s Equation.

Squaring the probability amplitude to find the probability of detection of a quantum particle in a particular state is called the “Born Rule.” It’s named after Max Born, who discovered this rule. Squaring the probability amplitude gets rid of the square root of -1 and so brings the output of the equation back into an understandable physical reality.

Notes:
1) Schrodinger’s Equation can be used to calculate the properties of other quantum particles besides electrons so long as they don’t travel too fast, that is, so long as they don’t approach the speed of light.
2) I don’t know if the term “probability amplitude” is used in other calculations in addition to its use in Schrodinger’s Wave Equation. For example, does it appear in Dirac’s Equation for particles traveling at or near the speed of light?
3) In the example of the hydrogen atom, I referred to a spherical orbital. Electrons in a hydrogen atom can have orbitals that are many different shapes. Each type of atom (carbon, lithium, oxygen, etc.) has its own set of orbitals that its electrons can occupy.