In the equation above, P(B\mid A) is usually called the likelihood. It represents the likelihood for B to be true given A is true. A can be a probability distribution that fits with the training data.
P(A) is usually called the prior, which is the probability for A to be true. For example, in a classification task, P(A) can be the prior probability of a sample, without any further information, belongs to a certain class.
P(B) is usually called the evidence. It is some piece of information that alters our judgement for how likely that A would be true.
P(A\mid B) is usually called the posterior probability. With evidence B being true, it is the probability for A to be true. For example, given a sample B, how likely it is from the distribution of A.