Short answers, pulled from the story.
A dynamic Bayesian network is a probabilistic graphical model that links variables across adjacent time steps. At any time T, the value of a variable is calculated from the current model structure and the immediately preceding state at time T-1, a structure known as the two-timeslice Bayesian network.
Dynamic Bayesian networks were developed by Paul Dagum in the early 1990s at Stanford University's Section on Medical Informatics. Dagum designed them to unify and extend older models such as Kalman filters, ARMA models, and hidden Markov models into a single general framework.
Dynamic Bayesian networks generalize hidden Markov models and Kalman filters, as well as linear state-space models and ARMA forecasting models. DBNs extend these tools to arbitrary nonlinear and non-normal time-dependent domains.
Dynamic Bayesian networks are commonly used in robotics and have been applied in speech recognition, digital forensics, protein sequencing, and bioinformatics. Researchers also use them for data mining and for modeling gene regulatory networks.
Several open-source tools support DBNs: Kevin Murphy's Bayes Net Toolbox for Matlab (GPL license), the Graphical Models Toolkit (GMTK) for rapid prototyping, libDAI (a C++ library under the FreeBSD license), aGrUM (C++ with Python bindings, GPLv3), and FALCON for regulatory network modeling in biology.
Dynamic Bayesian networks are conceptually related to probabilistic Boolean networks and can similarly be used to model dynamical systems at steady-state. Both frameworks reason about how system states evolve over time under uncertainty.