The Eigenstate
An eigenstate is a state that does not change under the operation of its defining operator. When you measure the observable associated with that operator, you get a definite value. No uncertainty. No probability distribution. A single number. This is what it means for a system to be "in a state" in quantum mechanics — a state that is an eigenstate of the observable you are looking at.
The mathematics is straightforward. An operator $\hat{A}$ acts on a state vector $|\psi\rangle$. If $\hat{A}|\psi\rangle = a|\psi\rangle$ for some number $a$, then $|\psi\rangle$ is an eigenvector of $\hat{A}$ with eigenvalue $a$. The state $|\psi\rangle$ is an eigenstate. The number $a$ is the value you will measure. If you measure $\hat{A}$ again immediately, you will get the same value. The state has been projected onto the eigenstate, and it stays there. This is the projection postulate. Measurement changes the state, then the state is what you measured.
For a Hermitian operator — any observable in quantum mechanics — the eigenvalues are real and the eigenvectors form a complete basis. Completeness means any state can be expanded as a superposition of eigenstates:
$$|\psi\rangle = \sum_n c_n|a_n\rangle$$
where $|a_n\rangle$ are the eigenstates and $c_n = \langle a_n|\psi\rangle$. The probability of measuring eigenvalue $a_n$ is $|c_n|^2$. The state $|\psi\rangle$ is not an eigenstate unless it equals a single $|a_n\rangle$ (up to a phase). A general state is a superposition. It does not have a definite value for the observable. It has a probability distribution. The eigenstate is the special case where the distribution collapses to a delta function.
This is what "being in a state" means operationally. A system is in a definite state of an observable when it is an eigenstate of that observable. Before measurement, it may not be. After measurement, it is. The act of measurement forces the system into an eigenstate. The act of time evolution (Schrödinger equation) may move it out again. The eigenstate is stable under its own operator but not necessarily under time evolution. An energy eigenstate is stable under time evolution. An eigenstate of position is not.
Position eigenstates $|x\rangle$ are eigenstates of the position operator $\hat{x}$: $\hat{x}|x\rangle = x|x\rangle$. They are delta functions in position space. They are not normalizable. They live in the rigged Hilbert space, not the Hilbert space. Momentum eigenstates $|p\rangle$ are plane waves: $\langle x|p\rangle = e^{ipx/\hbar}$. They are also not normalizable. They are idealizations. The physical states are wave packets — superpositions of momentum eigenstates that are square-integrable.
Energy eigenstates are normalizable when the spectrum is discrete. The particle in a box, the harmonic oscillator, the hydrogen atom — all have discrete energy eigenstates that are proper vectors in the Hilbert space. The free particle has a continuous energy spectrum. Its energy eigenstates are plane waves, not normalizable. The distinction between discrete and continuous spectrum matters for normalization. Discrete eigenstates have $\langle n|m\rangle = \delta_{nm}$. Continuous eigenstates have $\langle x|x'\rangle = \delta(x-x')$. The Kronecker delta versus the Dirac delta. Discrete sums versus integrals.
The concept of eigenstate is not specific to quantum mechanics. It appears in linear algebra, in differential equations, in any theory with linear operators. But in quantum mechanics, it has physical content. The eigenstate is not just a mathematical abstraction. It is a physical preparation. You prepare an eigenstate by measuring the corresponding observable and selecting the outcome. You filter. You isolate. An eigenstate is a prepared system.
Time evolution of an eigenstate of $\hat{H}$ is trivial: $|n(t)\rangle = e^{-iE_nt/\hbar}|n(0)\rangle$. The state picks up a phase. Physical predictions do not change. An energy eigenstate is stationary. An eigenstate of some other operator $\hat{A}$ that does not commute with $\hat{H}$ evolves non-trivially. The probabilities for measuring $\hat{A}$ oscillate in time. The eigenstate is only an eigenstate at the moment you prepared it.
Entanglement complicates the picture. For a composite system, the eigenstates of the total Hamiltonian may be entangled states — states that cannot be written as a product of subsystem states. The eigenstate is still well-defined. It is still a vector in the tensor product Hilbert space. But it does not factor. Measuring one subsystem does not leave the other in a definite eigenstate of its own Hamiltonian. The eigenstate of the whole is not an eigenstate of the parts. This is not a failure of the concept. It is the concept revealing its structure in a higher-dimensional space.
The eigenstate is the anchor of quantum measurement. Before measurement, the state is uncertain. The measurement outcome is probabilistic. After measurement, the state is an eigenstate of the measured observable, and the outcome is certain. The transition from probability to certainty is the projection postulate. The eigenstate is what certainty looks like in quantum mechanics. Everything else is superposition, and superposition is what makes quantum mechanics not classical.