(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 (generative) 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 in terms of inference, Active Inference models both perception and action as inference under closely related unifying objectives: whereas perception minimizes variational free energy (which sometimes reduces to precision-weighted prediction error), action minimizes expected free energy.
The Free Energy Principle is a generalisation of Active Inference that not only attempts to describe biological organisms, but also "things" that can be separated from their environments via a (Markov blanket) over some timescale.
External Links:
Book Review: Surfing Uncertainty - Introduction to predictive processing by Scott Alexander
Predictive Processing And Perceptual Control by Scott Alexander
Predictive coding under the free-energy principle by Karl Friston and Stefan Kiebel
Related Pages: Perceptual Control Theory, Neuroscience, Free Energy Principle