An adjectival phrase is one that describes or modifies a noun, and an adverbial phrase is one that modifies a verb. Sometimes these phrases exist with only a qualifier, which attributes certain qualities to other words, and either an adjective or an adverb.
An adjective phrase (or adjectival phrase) is a phrase the head of which is an adjective, e.g. fond of steak, very happy, quite upset about it, etc. The adjective can initiate the phrase (e.g. fond of steak), conclude the phrase (e.g. very happy), or appear in a medial position (e.g. quite upset about it).
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