Az alapítványnak van könyvtára.

Because Hungarian does not usually use a separate verb meaning to have the way English does. Instead, possession is often expressed with this pattern:

possessor + -nak/-nek + van + possessed thing

So alapítványnak means roughly to the foundation.
The whole sentence is literally something like:

To the foundation, there is its library.

That is how Hungarian says The foundation has a library.

Van is the 3rd person singular form of lenni (to be), so it literally means is / there is.

In possession sentences, Hungarian uses van where English uses has:

• A fiúnak van kutyája. = The boy has a dog.
• Az alapítványnak van könyvtára. = The foundation has a library.

So here van is the key word that creates the have meaning together with the dative possessor.

Because the possessed noun in Hungarian usually gets a possessive ending.

Here the owner is the foundation, which is a 3rd person singular possessor (his/her/its type), so könyvtár gets the 3rd person singular possessive suffix:

• könyvtár = library
• könyvtára = his/her/its library

So Hungarian marks possession on the possessed thing itself, not only on the owner.

Because the possessive suffix and definiteness are two different things in Hungarian.

• könyvtára shows whose library it is
• the presence or absence of an article shows whether it is definite or indefinite

So:

• van könyvtára = has a library
• van a könyvtára would mean something more like has its/the library, which is not the normal meaning here

In this sentence, the noun is bare, so it is understood as indefinite: a library.

Az is the definite article the.

It is used because alapítvány refers to a specific foundation, not just any foundation:

• Az alapítványnak... = The foundation...
• Egy alapítványnak... = A foundation...

So the sentence begins with a definite possessor: the foundation.

That is normal in Hungarian.

In this kind of sentence:

• the possessor is marked with -nak/-nek
• the possessed noun also gets a possessive suffix

So both parts help show the relationship.

This may feel redundant from an English point of view, but in Hungarian it is the standard pattern:

• A lánynak van testvére. = The girl has a sibling.
• Az alapítványnak van könyvtára. = The foundation has a library.

Because of vowel harmony.

Hungarian suffixes often have two forms, and the choice depends on the vowels in the word.
For alapítvány, the word takes the back-vowel form, so the suffix is -nak:

• -nak / -nek

Thus:

• alapítvány
  • -nak = alapítványnak

This is a very common pattern in Hungarian grammar.

Yes, but the word order changes the emphasis.

The neutral, ordinary sentence is:

Az alapítványnak van könyvtára.

If you say:

Az alapítványnak könyvtára van,

that puts more focus on könyvtára. It can sound like you are highlighting what the foundation has.

Hungarian word order is flexible, but different orders often signal different emphasis or information structure.

Because this is not the same kind of sentence.

Hungarian often omits van in simple equational sentences in the 3rd person present:

• A könyvtár nagy. = The library is big.
• Ő tanár. = He/She is a teacher.

But in existence and possession sentences, van is normally kept:

• Van egy könyvtár. = There is a library.
• Az alapítványnak van könyvtára. = The foundation has a library.

So here van cannot simply be dropped.

You would usually use nincs:

Az alapítványnak nincs könyvtára.

That means The foundation does not have a library.

This is the negative counterpart of van in this kind of sentence:

• van = there is / has
• nincs = there is not / does not have

Yes, you could, but it is slightly different in feel.

• Az alapítványnak van könyvtára. = a general statement that the foundation has a library
• Az alapítványnak van egy könyvtára. = the foundation has a/one library, often with a bit more emphasis on the existence of one such thing

In many cases, the version without egy is the more natural plain statement of possession.