You are referring to the Pauli exclusion principle.

Definition
The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously within a quantum system.

Formal statement
For particles with half-integer spin (fermions), the total quantum-mechanical wavefunction is antisymmetric under exchange of two identical particles. As a direct consequence, two fermions cannot share the same complete set of quantum numbers.

In atomic physics
In an atom, each electron is described by four quantum numbers:

• principal quantum number $n$

• orbital angular momentum $l$

• magnetic quantum number $m_l$

• spin quantum number $m_s$

No two electrons in the same atom can have all four quantum numbers identical. This is why each atomic orbital can contain at most two electrons, and only if their spins are opposite.

In solid-state physics
The principle explains:

• the electronic band structure of solids,

• why metals have conduction electrons,

• why insulators and semiconductors exist,

• the concept of a Fermi sea and Fermi energy.

In astrophysics
The Pauli exclusion principle gives rise to degeneracy pressure, which:

• prevents white dwarf stars from collapsing (electron degeneracy pressure),

• stabilizes neutron stars (neutron degeneracy pressure).

What it does not apply to

• It does not apply to bosons (integer-spin particles), which can occupy the same quantum state in unlimited numbers (e.g., photons in a laser).

Conceptual interpretation
The exclusion principle is not a classical force or repulsion. It is a fundamental quantum constraint arising from particle indistinguishability and the symmetry properties of the wavefunction.

If you want, I can also express it in operator language, in terms of creation–annihilation operators, or connect it explicitly to antisymmetrization in many-body quantum mechanics.