the posterior probability

The posterior is your belief AFTER the evidence has been taken into account. It is the output of Bayes' theorem, the P(A given B) on the left-hand side: the revised probability of the hypothesis once you fold in what you observed. If your prior that a friend has a disease was 0.5%, and then a test comes back positive, the posterior is the new, higher (or in surprising cases, still-low) probability you now hold.

The posterior is built from the other two parts: posterior is proportional to prior times likelihood, with the proportionality fixed by dividing by P(B) so everything sums to 1 across the hypotheses. The prior brings in what you knew; the likelihood brings in what you just saw; the posterior is their reconciliation. Crucially, the posterior is a complete answer only relative to the evidence you used — feed in more data later and the posterior becomes the new prior for the next update. This is what makes Bayesian reasoning sequential and self-correcting: today's posterior is tomorrow's prior.

Posteriors are what decisions actually rest on. A spam filter acts on the posterior probability that a message is spam, not on the raw word-frequencies. A diagnosis weighs the posterior probability of illness given all the tests. The recurring lesson is that a strong-looking piece of evidence can still leave the posterior modest if the prior was very low — the arithmetic of Bayes refuses to let a single test overrule a rare base rate.