It's not possible to answer this question without differentiating some very important ideas. There's a difference between a part of speech or a type of phrase and the job it does in its larger phrase or clause.
Parts of speech have names like noun, adjective, verb. These refer to types of word. These types of word have certain groups of properties within a given language. So, for example, English nouns usually have singular and plural forms. They are usually pre-modified by either adjectives or nouns and they are never pre-modified by adverbs. There are a whole load of other properties that nouns can have. If you'd like to find out what (some of these) these are, see here. English adjectives on the other hand do not have singular and plural forms. They do often have comparative forms, like bigger and biggest and so forth.
Types of phrase have names like noun phrase, verb phrase, adjective phrase and so forth. These describe chunks of words built around, for example, a noun, verb or adjective.
The jobs that such words, or chunks of words, happen to be doing in a larger phrase or clause are often called syntactic functions or grammatical relations. These have names like Subject, Object, Locative Complement and Determiner. I will adopt the practice of using initial Capital Letters when talking about grammatical relations, as I have done in this paragraph.
Determiners within noun phrases.
Noun phrases come in two sections, in the same way that clauses do. While clauses have a Subject and a Predicate, noun phrases have a Determiner and a Head (some grammars may use different words for these). Note that these words refer to the jobs that different types of phrase may carry out within a given phrase or clause, not to the type of word or phrase that is carrying out that job.
Here are some examples of noun phrases with the Determiner placed in brackets. Note that adjective appears in the Head, and not in the Determiner:
- [The] big elephant.
- [That] big elephant.
- [My wife's] big elephant.
- [My wife's sister's] big elephant.
- [The woman you met yesterday's] big elephant.
- [Her] big elephant.