In the three examples above, for people who use the term in a principled way, the phrases very well, on Saturdays and every day are Adverbials. They are phrases which have a special function in the clause structure. The term adverbial is derived from the old-fashioned and unobservant(and not very acutely observant) idea that different parts of speech are related to specific different syntactic functions or grammatical relations. Adverbs have traditionally been regarded as having the special function of modifying verbs or verb phrases. The term adverbial, therefore, is used to describe phrases which have the function of modifying verbs, verb phrases or clauses. Remember though that there are many different types of phrase that can modify verb phrases, not only adverbs. The term adverbial then, when used in this way, means something like having an adverbish type of syntactic function. It does not mean that the phrase is an adverb or anything like that.