the stationary (invariant) distribution

Suppose you found a way to spread probability over the states so that, after one step of the chain, the spread looks exactly the same as before. Nothing flows on net; the picture is frozen even though individual tokens keep moving. That self-reproducing distribution is the stationary distribution — the chain's equilibrium, the way it 'wants' to be distributed once it has run for a long time.

A row vector pi (with non-negative entries summing to 1) is stationary if pi P = pi: applying one step leaves it unchanged. Writing it out, pi(j) = sum over all i of pi(i) P(i, j) for every j — the probability flowing INTO each state j exactly equals the probability already there. To find pi you solve this linear system: the balance equations pi P = pi together with the normalisation sum of pi(i) = 1. It is an eigenvector problem — pi is the left eigenvector of P for eigenvalue 1 — that small linear algebra cracks open.

For an irreducible positive-recurrent chain the stationary distribution exists and is unique, and pi(i) equals one over the expected return time to i — states you return to quickly carry more equilibrium weight. Crucially, stationary is NOT the same as limiting: every limiting distribution is stationary, but a periodic chain has a perfectly good stationary pi it never converges to. The stationary distribution is the backbone of PageRank, of queueing analysis, and of every Markov chain Monte Carlo method that engineers a chain whose equilibrium is the distribution you wish to sample.