Definition
Bayesian probability treats probability as a numerical measure of a rational agent's degree of belief in a proposition, on a scale where 0 means certain falsity and 1 means certain truth. Beliefs are updated by evidence using Bayes' theorem: the posterior probability is proportional to the prior multiplied by the likelihood of the observed data.
Unlike the frequentist interpretation, Bayesianism happily assigns probabilities to one-off events ('it will rain tomorrow', 'this defendant is guilty', 'this theory is correct'), because beliefs need not be tied to a repeatable experiment.
Why it matters
How it works
A Bayesian starts with a prior distribution that encodes belief before any data is seen. When data arrives, the likelihood of that data under each possible hypothesis is computed. Multiplying prior by likelihood (and renormalising) gives the posterior — the updated belief. The posterior becomes the prior for the next round of evidence, and the cycle continues.
The mechanics are just arithmetic with conditional probabilities, but the conceptual leap is treating belief itself as a probability. This view, neglected for much of the 20th century, has surged back to prominence with the computational tools (MCMC, variational inference) that make Bayesian calculations practical.