Probabilistic programming uses computer science techniques to do automated statistical modeling. For example, imagine I have a coin, and I want to discover if it is biased, i.e. if it lands on heads more often than tails. In a probabilistic programming framework, I can express my model as a simple Bernoulli model, `x ~ Bernoulli(p)`, and then automatically estimate the bias parameter `p` given some data (do "inference").
You can easily do this calculation by hand or in Python, but this does not generalize to more complex real-world scenarios. For complex probabilistic models, we must rely on numerical approximations. MCMC is just one algorithm for doing this approximate inference. Another popular technique is called variational inference [2]. Another commenter mentioned HMC [3], which is just a specific instance of MCMC.