(From Wikipedia) predictive processing (a.k.a. predictive coding, the Bayesian Brain hypothesis) is a theory of brain function in which the brain is constantly generating and updating a mental model of the environment. The model is used to generate predictions of sensory input that are compared to actual sensory input. This comparison results in prediction errors that are then used to update and revise the mental model.
Active Inference can be seen as a generalisation of predictive processing. While predictive processing only explains the agent's perception, Active Inference models both perception and action as inference under the same unifying objective: optimisation of the informational quantity called variational free energy.