The forward-backward algorithm is an algorithm for computing posterior marginals in a hidden Markov model (HMM). It is based on dynamic programming, and has linear complexity in the length of the sequence. It is used as a component of several other algorithms, such as the Baum_Welch algorithm and block Gibbs sampling in factorial HMMs. <read on>
Belief propagation is a message-passing algorithm for performing inference on graphical models, such as Bayesian networks and Markov random fields. It calculates the marginal distribution for each unobserved node, conditional on any observed nodes. Belief propagation is commonly used in artificial intelligence and information theory and has demonstrated empirical success in numerous applications including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability.
The algorithm was first proposed by Judea Pearl in 1982, who formulated it as an exact inference algorithm on trees, which was later extended to polytrees. While it is not exact on general graphs anymore, it has been shown to be a useful approximate algorithm.