stationary distribution
A stationary distribution is the resting point of a random process: if the chain is already in this distribution, taking one more step leaves it unchanged. It is the long-run fraction of time the system spends in each state, regardless of where it began (for nice chains).
Formally, for a Markov matrix P it is a probability vector pi with pi = P pi (column convention) and entries summing to 1. So pi is an eigenvector of P for eigenvalue 1. In the row convention it is the left eigenvector: pi^T = pi^T P. Solving (P - I) pi = 0 together with the normalization sum(pi) = 1 pins it down.
Existence is guaranteed because 1 is always an eigenvalue of a stochastic matrix. Uniqueness and convergence from any start come from Perron-Frobenius: if the chain is irreducible (every state reachable from every other) and aperiodic, then pi is unique and P^k p_0 -> pi for any starting p_0.
Why it matters: the stationary distribution is what PageRank computes, what tells you the equilibrium occupancy of a queue, and what makes Markov chain Monte Carlo work. The caveat: a periodic chain has a stationary distribution but never settles into it (it keeps cycling), and a reducible chain can have many stationary distributions depending on which trap you fall into.
The stationary vector is the eigenvector for eigenvalue 1, normalized to sum to one.
Stationary is not the same as reversible. A stationary distribution just satisfies pi = P pi; a reversible chain additionally satisfies detailed balance pi_i P_ji = pi_j P_ij, a stronger symmetry that many sampling algorithms exploit.