Solve pi = pi P with the entries summing to 1. It answers "in the long run, what fraction of time is spent in each state".
Existence and uniqueness require the chain to be irreducible and aperiodic. A periodic chain has a stationary distribution but does not converge to it - it cycles, which is a distinction interviewers occasionally probe.
The shortcut worth knowing: for a random walk on an undirected graph, the stationary probability of a vertex is proportional to its degree. That collapses some intimidating questions into one line.