In a Markov chain, states are classified as either recurrent or transient based on the probability of returning to them after leaving. A recurrent state is one where there is a 100% probability of eventually returning to it, no matter how far into the future, implying it is part of a closed set of states that the chain will cycle through. In contrast, a transient state is one where there is a nonzero probability of never returning once it is left, meaning the chain can escape from it permanently. Recurrent states are often part of stable long-term behavior, while transient states are temporary and may only be visited during the chain's early stages.