Robert and Casella (2013) provides the following definition:

For illustration, you can visit the jupyter notebook on MCMC_example

Working principle: For an arbitrary starting value x(0)x^{(0)}, a chain X(t)X^{(t)} is generated using a transition kernel with stationary distribution ff, which ensures the convergence in distribution of X(t)X^{(t)} to a random variable from ff.

Some good materials about MCMC.