Short answers, pulled from the story.
A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph. This structure uses nodes to represent variables in the Bayesian sense, which may be observable quantities, latent variables, unknown parameters or hypotheses.
Judea Pearl coined the term Bayesian network in 1985 to emphasize subjective input information and reliance on Bayes conditioning for updating knowledge. His book Probabilistic Reasoning in Intelligent Systems appeared in the late 1980s alongside Neapolitan's Probabilistic Reasoning in Expert Systems.
Cooper proved exact inference in Bayesian networks is NP-hard while working at Stanford University on large bioinformatic applications in 1990. Dagum and Luby proved in 1993 that no tractable deterministic algorithm can approximate probabilistic inference within an absolute error less than one half.
Notable software packages include OpenBUGS which serves as open-source development of WinBUGS and SPSS Modeler offers commercial implementation capabilities for building Bayesian networks within enterprise environments. Stan provides an open-source package using No-U-Turn sampler variants of Hamiltonian Monte Carlo methods.