An adverbial clause is a clause that functions as an adverb. In other words, it contains a subject (explicit or implied) and a predicate, and it modifies a verb.
I saw Joe 'when I went to the store. (explicit subject I)
He sat quietly in order to appear polite. (implied subject he)
According to Sidney Greenbaum and Randolph Quirk, adverbial clauses function mainly as adjuncts or disjuncts. In these functions they are like adverbial phrases, but due to their potentiality for greater explicitness, they are more often like prepositional phrases (Greenbaum and Quirk,1990):
We left 'after the speeches ended. (Is it an adverbial clause, adverbial phrase, or prepositional phrase?)
We left after the end of the speeches.(Is it an adverbial clause, adverbial phrase, or prepositional phrase?)
Adverb clauses are things that modifiy verbs adjectives and other adverbs. For example:
Hardly had I reached the station when the train started to leave the platform.
The adverb clause in this sentence is " when the train started to leave the platform " because it is subordinate clause and because it has the trigger word when which has no relation to the speaker but what happened as the train left.
Kinds of adverbial clauses
Greenbaum, Sidney & Quirk, Randolph. A Student's......