A stationary distribution in a Markov chain is a probability distribution over states that remains unchanged after the application of the transition probabilities. In other words, when the system reaches a stationary distribution, the probabilities of being in each state stabilize, meaning that if the system starts in this distribution, it will stay in the same distribution over time. This concept is important for analyzing the long-term behavior of a Markov chain because it describes the equilibrium state that the system approaches after a large number of transitions. The stationary distribution can provide insights into the steady-state probabilities of various outcomes, helping to predict the long-term trends of the system, such as the proportion of time spent in each state.