| This article is part of the grammar course.
You may choose to follow it in a structured way, or read each item separately.
The Adverbial, as its name suggests, does the job of an adverb. In form it may be an adverb (one word), an adverb phrase (most often in the form of a Prepositional Phrase, therefore with more than one word), or an Adverbial Clause - a group of words centred on a verb.
Examples: "he spoke quickly" (adverb in an intransitive Clause); "he thought in a very personal way" (Adverbial Phrase, intransitive sentence); "he sang while they were waiting" (Adverbial Clause, intransitive sentence); "John cooked the dinner in the oven" (Adverbial Phrase, transitive sentence); "Peter sang the song badly" (Adverbial, transitive sentence); "we are making hay while the sun shines" (Adverbial Clause, transitive Sentence); etc.
This is the most optional of all the elements of a clause. A writer can include it or not, with great freedom of choice. It is also more 'mobile' than the others - that is, it can appear in any part of the sentence: "While they were waiting, he sang."
One warning for beginners in grammar: sometimes Adverbials can look like Objects. Indeed, one type of Complement is the Adverbial Complement, e.g. "it is in the book"; "it was on Tuesday"; "it is growing out of all proportion". Here the Adverbial completes the meaning of a copular verb.