How specifically do I need to describe (in words) the language of a regular expression? For the regular expression $(a^*ba^*b)^*a^*$, I want to learn how to describe it in words. I know there are an even number of $b$s but how much do I need to explain it? Do I need to explain it enough that each possible string can be produced from my explanation?
Another example is $(a^*[ba^*c])^*$ where I would describe it as as string over $\{a,b,c\}$ where there are the same number of $a$s and $c$s.
The first language can be described as the set of words on the alphabet $\{a,b\}$ with an even number of $b$'s.

The second one is not the language of words on the alphabet $\{a,b,c\}$ with an equal number of $b$'s and $c$'s (this condition would define a non-regular language). You could use greybeard's description or say: the language contains the empty word, the last letter of the nonempty words of the language is a $c$ and if you delete the $a$'s, you are in $(bc)^*$.

It would be safe to provide a formal proof by computing the minimal automaton corresponding to each description.