I'll try a general explanation of terms: see if it answers your specfic questions. If not, come back to me:
The terms 'adjective' and 'adverb' refer to single words such, respectively, as 'red' and 'quickly'.
An adjective phrase is a group of words headed by an adjective. By 'headed' is meant that that is the syntactic focal point, so to speak - the word to which the others in the phrase stand merely as secondary components, or 'adjuncts'. Thus 'very big' is an adjective phrase: it consists of a head - the adjective 'big' - modified by the adverb 'very'.
A 'phrasal (or 'compound') adjective', on the other hand, is a group of words not headed by a true adjective, but which nevertheless serve together to modify a noun. An example would be 'big-hearted' - there is no single adjectival head (as there is no individual word 'hearted'), simply an indivisible, nonheaded phrase but one which nevertheless serves an adjectival (or 'adnominal') function.
Similarly we have 'adverb phrases', such as 'extremely quickly' (= head 'quickly' modified by adverb 'extremely').
A prepositional phrase is a nonheaded phrase-type consisting of a preposition plus an object noun or pronoun, and normally serves either an adverbial or an adjectival (adnominal) function, such as respectively 'in the room' in 'they sat in the room' and 'with the ugly wife' in 'the man with the ugly wife'.
Similarly again we have the 'noun phrase', but this is used with two slightly differing senses: in the strictest sense, it is any phrase headed by a noun (as 'the tall boy' = head 'boy' modified by adjective 'tall' and determined by article 'the'). In a looser sense (especially in Chomskyan linguistics), however, it denotes any phrase serving like an noun, and can thus include not only a simple noun or pronoun but even an infinitive phrase functioning as sentence-subject, such as 'to study' in 'to study is a right, not a duty'. Another, somewhat more common, term for 'noun phrase' in this very wide sense is a NOMINAL.
Similarly, we have the terms ADNOMINAL to denote any word/phrase relating to a noun, ranging from a simple adjective to a relative clause, and ADVERBIAL to denote any word/phrase that serves like an adverb (up to and including adverbial clauses).
[Beware, though, that some grammarians - notably Quirk and Greenbaum - prefer to restrict the term 'adverbial' to words/phrases modifying main verbs or standing as complements, thus excluding single-word adverbs such as 'very' or 'extremely' that occur only as adjuncts in adjective or adverb phrases. Although I respect their right to do so - since they do it clearly and explicitly - it is a restriction that, I must confess, I consider a rather unnecessary complication in an already complex terminological area, and therefore do not myself apply!]
Note that all of the above are terms denoting either word-classes (single words) or function-types (words or phrases corresponding functionally to particular word-classes).
Additionally we have what might be considered 'higher-level' function types that subsume the typical functions of more than one word-class: DETERMINERS (which can be articles or adjectives), MODIFIERS (= optional - and, therefore, non-complemental - adnominals or adverbials) and CONNECTORS (conjunctions and other words serving to connect clauses or sentences: please see my answer to your post 'conjuncts' for a detailed explanation of these).
Then we have terms denoting grammatical ROLES: subject, object and complement. These are distinguished in that the first two can be served only by nominals, while the third (as explained in another recent post) can be any of various forms that serve to complete syntactically or semantically a preceding verbal, nominal, adnominal or adverbial element.
A clause is divided into two main parts, a subject (also, as stated, a 'role') and a predicate (everything that is not either the subject or an adjunct to the subject). The predicate itself can be further subdivided into PREDICATOR (the finite verb component of its main verb phrase) and PREDICATION (the rest of the predicate). Thus the sentence 'The boy is walking along the road' could be analysed as subject (the boy) + predicator (is) + predication (walking along the road).
Finally, regarding the term 'verb phrase' (a phrase headed by a verb form), note that
(1) this will normally be taken to include direct objects as well as adverbials, both classifiable as adjuncts to the verb;
(2) corresponding to the extended sense of 'noun phrase' noted above, Chomskyan linguistics uses 'verb phrase' as a synonym of 'predicate', so that any sentence, in Chomskyan terms, is analysable as a combination [NP + VP].
(3) A verb phrase, even in its strictest sense, can constitute an entire sentence if the predicator is a form of the imperative mood (as the sentence 'Go!');
(4) A verb phrase may be either FINITE (i.e. headed by a 'main' verb - one inflectable as to mood, tense, person and number) or NONFINITE (an infinitive or participial phrase - typically serving a nominal, adnominal or adverbial role).
and regarding the term 'clause' that, classically, this must consist of a subject plus a finite verb (i.e. the essential elements of a sentence), but that many grammarians these days apply the term 'nonfinite clause' to phrases headed by nonfinite verb forms, such as participial phrases.
I hope that these notes will go some way toward clarifying these issues for you!