There are different kinds of MCMC algorithms: Metropolis-Hastings, Gibbs, importance/rejection sampling (related).
- Metropolis-Hastings
- Gibbs
- Importance/rejection sampling (related).
Why would one use Gibbs sampling instead of Metropolis-Hastings? I suspect there are cases when inference is more tractable with Gibbs sampling than with Metropolis-Hastings, but I am not clear on the specifics.