What is actually happening when a chemical bond is formed? Just accepted that it was a thing while in school so I could get through tests and homework in chemistry but now I'm working through a book on semiconductor device physics and I can't accept not understanding what is actually going anymore.
 A: First of all, let's go through some basics,
Orbitals: Orbitals are the mathematical functions that describes the wave-like behavior of either one electron or a pair of electrons in an atom. Orbitals can be considered as "electron clouds" that exist around each nucleus.
Wave function (Ψ): As per the de Broglie principle, every particle can be considered as a wave. The electron is no exception. A mathematical function that describes the wave nature of an electron in an atom is called the wave function.
The wave function of an orbital can be obtained by solving the Schrodinger's equation (for hydrogen like ions):
$$\frac{-h^2}{8\pi m}\left[\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2}\right]-\frac{Ze^2\psi}{4\pi\epsilon_0r}=E\psi$$
where $x$, $y$, and $z$ refer to the coordinates of a point, $\psi$ refers to the amplitude of the wave function at $(x,y,z)$, $Z$ refers to the atomic number (number of protons), and $E$ refers to the energy of the orbital. $E$ can be found out by:
$$E=\frac{mZ^2e^4}{8\epsilon_0^2h^2n^2}$$
Keep these basics in mind.

Remember how we considered the electron to be a wave? That means that electrons follow the properties of waves, such as interference and diffraction. In such cases, the electron waves reinforce together (constructive interference) when they have the same sign for $\psi$, and cancel each other out (destructive interference) when they have opposite sign for $\psi$
Let's get back to your question "How does a bond form". What is a bond in the first place? It is the force of attraction between two atoms that keeps them together. There are many types of bonds, although here, I'll be discussing about covalent bonds.
For simplicity, lets consider two $\mathrm{p}$-orbitals overlapping head-on to form a $\sigma$-bond.
The $\mathrm{p}$-orbitals have two lobes, with one of them having a +ve value for $\psi$, and the other lobe having a -ve value of $\psi$. For simplicity, I'll consider the +ve lobe with an "up arrow" and the -ve lobe with a "down arrow"

Case 1: Constructive interference
Lets say both "up arrow" sides overlapped each other. This would cause a constructive interference of orbitals, and increase the electron density between the nuclei.

The electron cloud pulls the two nuclei together, and balances out the internuclear repulsions. This is a stable bond, and the orbital formed due to the addition is called a bonding orbital.
**Case 2: Destructive interference
Lets say this time the "up arrow" overlapped with the "down arrow". This would cancell out the electron density in the middle, and increase it on the edges.

Here, both the nuclei repel each other, and the outside electron clouds promote this by further pulling it away. This is the case of an unstable antibond, and the orbital formed is called an antibonding orbital.

This should be a sufficient enough information. Drop a comment if you need anything else.
