Adverbial clauses are clauses that modify verbs. For example, in the sentence,
Sarah cleaned up the mess while Harry had another drink.
the adverbial clause "while Harry had another drink" modifies the verb "cleaned up." And in the sentence
Harry was angry because Sarah had refused his offer.
"because Sara refused his offer" is an adverbial clause modifying the verb "be."
The two examples just given are finite adverbial clauses, but adverbial clauses are often non-finite. For example in the sentence
After refusing Harry's offer, Sarah left immediately.
the non-finite adverbial clause, "after refusing Harry's offer" modifies the verb "left."
Adverbial clauses are usually introduced by "subordinating conjunctions" such as:
when, while, before, after, because, although, if, unless