Question

What is the part of speech in each sentence? Is my explanation right?

1. This book is for you.
['For you' is an adverb phrase.]

2. The gift for you is a book.
['For you' is an "adjective" phrase.]

3. I bought this book for you.
['For you' is an adverb phrase.]

Answer 1

There are eight parts of speech in English.
https://en.wikipedia.org/wiki/Part_of_speech#Western_tradition

You are referring to each prepositional phrase's usage, not the part of speech.

Sentence 1 --
This = pronoun
book = noun
is = verb
for = preposition
you = pronoun

A prepositional phrase is comprised of a preposition plus its object (which can be either a noun or a pronoun). There is often an article and/or adjective between the preposition and its object.

Please revise your original statement. You're not referring to parts of speech here, but to the use of prepositional phrases: adjectival or adverbial.