an eigenstate

Among all the states an operator could act on, a few special ones survive its action almost unchanged — the operator merely rescales them. These are the eigenstates, and they are the states of definite value. An eigenstate of the energy operator is a state with one sharp energy; an eigenstate of spin-z is a state that is definitely up or definitely down. They are the pure tones out of which every other state is composed.

The defining relation is A|a> = a|a>: acting with the operator A on the eigenstate |a> returns the same state multiplied by a number a, the eigenvalue. Physically, if a system is in the eigenstate |a> of an observable A, then measuring A yields the value a with certainty (no spread). Because a Hermitian operator's eigenstates form a complete orthonormal basis, any state can be expanded as |psi> = sum_n c_n |a_n>, and the Born rule reads the probabilities |c_n|^2 straight off that expansion.

An important honesty about idealizations: for observables with continuous spectra, the eigenstates are not normalizable and hence are not genuine physical states. Momentum eigenstates are infinite plane waves e^{ipx/hbar}, and position eigenstates are Dirac delta functions — both are useful mathematical fictions (they satisfy a delta-function normalization) that you superpose to build real, normalizable wave packets. Bound-state energy eigenstates, by contrast, are honest normalizable states you can actually occupy.