The Gibbs sampler is a simple form of Markov Chain Monte Carlo simulation, widely used in Bayesian statistics, based on sampling from full conditional distributions for each variable or group of variables. The name comes from the method being first used on Gibbs random fields modeling of images by Geman and Geman (1984).
The Gibbs sampler is a type of Markov Chain Monte Carlo (MCMC) simulation that produces a Markov chain guaranteed to converge to a target distribution $\pi$ of interest.
Assuming this target $\pi$ is defined on a space of dimension larger than one, the sampler is based on iterated simulations from the full conditional distributions associated with $\pi$ for each variable, though variants exist, such as sampling from blocks of variables conditional on all other variables. There also exist versions for univariate targets $\pi$ based on completion schemes like slice sampling.
On convergence, each full iteration across all variables yields samples from the joint multivariate distribution. As with most MCMC schemes, successive iterates are generally dependent.
It is widely used in Bayesian inference, though it's not limited to Bayesian approaches. For instance, the popular BUGS software takes its acronym from Gibbs sampling, as it stands for Bayes Using Gibbs Sampling.