Kalman Filter: Exact belief update using Bayes rule for linear systems.
Extended Kalman Filter: Locally linearize the system and use Bayes rule.
Particle Filter: It is essentially a Monte Carlo simulation based method meant to be used when the posterior can not be computed exactly. You maintain a set of particles (sampled from the prior), with associated weights (probabilities) , and compute the posterior particles corresponding to them using a simulator. The posterior belief is then an expectation of the posterior particles.