A Bayesian update or belief revision is a change in probabilistic beliefs after gaining new knowledge. For example, after observing a patient's test result, we might revise our probability that a patient has a certain disease. If this belief revision obeys Bayes's Rule, then it is called Bayesian.
Bayesian belief updates have a number of other interesting properties, and exemplify many key principles of clear reasoning or rationality. Mapping Bayes's Rule onto real-life problems of encountering new evidence allows us to reproduce many intuitive features that have been suggested for "how to revise beliefs in the face of new evidence".