What does the posterior probability of a variable mean in the Bayes' rule?
The word posterior stems from the latin word posterior (base form posterus), which means later, after, behind.

In the context of probability and Bayes' theorem it denotes a probability, often written as a symbol $P(A|B)$, where $A$ and $B$ are random variables. This symbol $P(A|B)$ means the probability of $A$ after you have observed $B$. Hence the naming posterior probability, or short posterior.

In this context the probability of observing a certain value of the random variable $B$ is denoted by $P(B)$ and is called prior probability, or just prior for short sometimes. The word prior also stems from a Latin word, pri, which means before. The name hints that you will gather observations of $B$ before you calculate $P(A|B)$.

In this way you use prior information to improve your model of $A$, expressed by the posterior probability $P(A|B)$. By this your probability of $A$ is based on the observed outcome of $B$. Like it has been said by John Maynard Keynes "When the facts change, I change my mind. What do you do, sir?".