stationary state
A stationary state is a quantum state with a single, sharply defined energy, whose observable properties do not change with time. Although its wavefunction does keep evolving, it does so only by spinning its overall phase round and round; the probability distribution you would actually measure stays frozen. Drop an electron into a stationary state and its cloud of likely positions just sits there.
These states are the solutions of the time-independent Schrödinger equation, the standing waves of the quantum world. Like the fixed vibration patterns of a guitar string, each one has a definite frequency, and that frequency is the energy divided by Planck's constant. The lowest such state is the ground state; the rest are excited states stacked above it.
Stationary states matter because they are stable resting points and because they form a complete toolkit. Any state at all, however complicated and however much it changes in time, can be written as a combination of stationary states. Their unchanging individual behaviour is exactly what makes them the natural reference frame for understanding everything that does change.
Only an overall phase turns; the measurable probability density stands still.
A superposition of two stationary states with different energies is not itself stationary: the relative phase between them turns, so its probability distribution genuinely oscillates in time. Only a single energy gives a truly frozen distribution.