a stationary state
A stationary state is a quantum state of perfectly definite energy. Its name captures the striking fact that although the wavefunction keeps rotating in the complex plane, everything you could actually measure about it stands still: the probability cloud does not move, the average position does not drift, the energy is sharp and unchanging. It is the quantum analogue of a pure musical note.
Precisely, a stationary state is an eigenstate of the Hamiltonian, H|psi_n> = E_n|psi_n>. Its full time dependence is only an overall phase, Psi_n(x, t) = psi_n(x) e^{-i E_n t / hbar}. When you form the probability density that phase cancels, |Psi_n|^2 = |psi_n|^2, which is time-independent — hence stationary. More generally the expectation value <A> of any operator that itself does not depend on time is constant in a stationary state.
Two honest cautions. First, stationary does not mean nothing happens: a stationary state can carry a steady, nonzero probability current (an electron orbital is a static cloud yet can carry circulating current). Second, and crucially, a superposition of two stationary states with different energies is not stationary — the relative phase e^{-i(E_2 - E_1)t/hbar} survives, so |Psi|^2 oscillates at the Bohr frequency (E_2 - E_1)/h. That beating is precisely what makes atoms radiate and dynamics happen at all.
A hydrogen atom sitting in its 2p orbital is in a stationary state: its charge cloud has a fixed shape in time. Only a superposition of, say, 1s and 2p oscillates and can radiate a photon at frequency (E_2p - E_1s)/h.
One energy eigenstate is frozen; mixing two different energies makes the atom oscillate and shine.
A common error is thinking any static-looking wavefunction is stationary; stationarity is specifically being an eigenstate of H, and superposing different energies destroys it.