A Markov chain, also sometimes referred to as Markov process, is a specific type of a stochastic process in which the evolution (i.e. the next configuration) only depends on the current configuration of the system. In other words, for each step of the model, all the probabilities of transitions that relate the different possible outcome configurations only depend on the current configuration: there is no "memory" of what led the system in the current configuration that is propagated to the new probabilities.
There are slightly different technical definitions for Markov processes when they are defined on discrete versus continuous variables, and with discrete or continuous parameters (i.e. time). For example a discrete value / discrete parameter example is the random walk on a lattice, while its continuous counterpart is the Wiener process that describes Brownian motion. Another example is the Gambler's ruin, a model that is used to describe how certain betting strategies are a mathematically guaranteed loss.
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