time-independent Schrödinger equation
The time-independent Schrödinger equation is what you get when you ask not how a system changes, but which energies it is allowed to have. It reads Ĥψ = Eψ: the Hamiltonian acting on a wavefunction simply returns the same wavefunction multiplied by a number E, the energy. This is an eigenvalue problem, and only special wavefunctions, with their special energies, satisfy it.
These special solutions are the stationary states, the standing-wave shapes of a quantum system. For a given potential — the walls of a box, the pull of a nucleus, the bowl of a harmonic oscillator — solving this equation tells you the complete menu of allowed energies and the wavefunction that goes with each. For bound systems that menu is discrete, which is the mathematical origin of quantised energy levels.
It is not a separate law from the full equation but a powerful special case. When the Hamiltonian does not change with time, you can peel the time dependence off into a simple oscillating phase, and what remains is exactly this equation. Solve it once and you have the building blocks from which every time evolution can be assembled.
An eigenvalue equation whose solutions are the allowed energies E and their stationary shapes ψ.
The name can mislead: the wavefunctions it gives still evolve in time, but only by an overall phase that does not affect any probability. The equation is 'time-independent' only because the Hamiltonian is, not because nothing changes.