Monte Carlo integration is a technique for numerically integrating a function by evaluating it at many randomly chosen points. It's useful for computing integrals when a closed form solution doesn't exist, and when the problem is high dimensional (in this case, standard numerical integration methods based on quadtrature are inefficient). The function to be integrated need not be a probability distribution.
Markov chain Monte Carlo (MCMC) refers to a class of methods for sampling from a probability distribution. It works by constructing a Markov chain whose equilibrium distribution matches the distribution of interest, then sampling from the Markov chain. This is useful when one cannot directly sample from the distribution of interest, particularly in high dimensional settings. Gibbs sampling is an MCMC method.
Monte Carlo integration and MCMC both fall under the general category of Monte Carlo methods, which use random sampling (the name refers to the Monte Carlo casino in Monaco). But, as above, they're used for completely different purposes (integrating a general function vs. sampling from a probability distribution).
A connection arises when MCMC methods are used for inference. For example, suppose we want to estimate a parameter as the mean of the posterior distribution. We can use MCMC to sample from the posterior, then take the mean of the samples. This corresponds to a form of Monte Carlo integration over the posterior.
