We know that adjective is a word which qualifies noun. A phrase which is used as an adjective is called as Adjective Phrase. That is, this phrase does the work of an adjective.
a. The girl wore a dress of cotton.
b. I purchased a bicycle with a red basket.
In sentence ‘a’, the phrase ‘of cotton’ describes the noun ‘dress’. So ‘of cotton’ is an adjective phrase. It is made of ‘of’ (preposition) and ‘cotton’ (noun).
In sentence ‘b’, the phrase ‘with a red basket’ is an adjective phrase. It is made of ‘with’ (preposition), ‘a’ (article), ‘red’ (adjective) and ‘basket’ (noun).
It does not matter that what the phrase is made up of but its work is important in a clause or sentence.
Keep in mind that, present participle of verb also can be used as an adjective and also can be used to make a phrase.
a. I like to see a smiling face.
b. A girl smiling with a joy, met me.
In sentence ‘a’, ‘smiling’ is used as an adjective for ‘face’.
In sentence ‘b’, ‘smiling with a joy’ does not have subject and predicate. So it is a phrase. It qualifies the noun ‘girl’. So it is an adjective phrase that contains present participle of the verb ‘smile’.