Discuss all adverbial clauses and phrases with good examples illustrated in meaningful sentences?

An adverbial clause is a dependent 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. (adverbial clause)
We left after the end of the speeches. (adverbial prepositional phrase)

Contrast adverbial clauses with adverbial phrases, which do not contain a clause.
I like to fly kites for fun.

Adverbial clauses modify verbs, adjectives or other adverbs. For example:
Hardly had I reached the station when the train started to leave the platform.

The adverbial clause in this sentence is "when the train started to leave the platform" because it is a subordinate clause and because it has the trigger word (subordinate conjunction) "when".