Ő a könyvtárban tanul.

Yes. Hungarian often drops subject pronouns because the verb ending usually already shows the person and number.

So:

• Ő a könyvtárban tanul. = He/She is studying in the library.
• A könyvtárban tanul. = also He/She is studying in the library.

Including ő can make the subject clearer or more emphatic, especially if you want to contrast it with someone else:

• Ő a könyvtárban tanul, én pedig otthon.
  He/She studies in the library, and I study at home.

Hungarian ő can mean both he and she.

It can mean either he or she. Hungarian does not normally distinguish gender in third-person singular pronouns.

So ő may refer to:

• he
• she

You usually understand which one is meant from context.

Here a is the definite article, meaning the.

• a könyvtár = the library
• könyvtár = library / a library depending on context

Hungarian has two forms of the definite article:

• a
• az

Which one is used depends on the sound that follows. Before a consonant, Hungarian usually uses a:

• a könyvtár

Before a vowel, it usually uses az:

• az iskola = the school

Hungarian often adds endings directly to nouns instead of using separate prepositions.

• könyvtár = library
• könyvtárban = in the library

The ending -ban / -ben means in.

So:

• a könyvtárban = in the library

This is one of the Hungarian case endings. Instead of saying a separate word for in, Hungarian attaches the idea to the noun.

Hungarian vowel harmony determines which form of the ending is used.

The ending meaning in has two common forms:

• -ban
• -ben

Words with mostly back vowels usually take -ban.
Words with mostly front vowels usually take -ben.

• könyvtár has back vowels, so: könyvtárban
• szék would take -ben: székben = in the chair

So the choice is based on the vowel pattern of the word, not on meaning.

The base form is könyvtár, which means library.

You can break könyvtárban into:

• könyvtár = library
• -ban = in

So:

• könyvtárban = in a/the library
• a könyvtárban = in the library

Tanul means study or learn, depending on context.

In this sentence, it most naturally means:

• studies
• is studying

Examples:

• Magyarul tanul. = He/She is learning Hungarian.
• A szobában tanul. = He/She is studying in the room.

So tanul can refer either to the activity of learning something or to studying as an academic activity.

Because Hungarian does not mark gender in the verb.

English has:

• he studies
• she studies

Hungarian uses the same verb form for both:

• ő tanul = he studies / she studies

The verb agrees with person and number, but not gender.

Yes. Tanul is present tense, and in many contexts Hungarian present tense can correspond to either:

• studies
• is studying

So Ő a könyvtárban tanul can mean:

• He/She studies in the library.
• He/She is studying in the library.

The exact meaning depends on context. Hungarian does not always distinguish simple present and present continuous the way English does.

Yes, Hungarian word order is flexible, and changing it often changes focus or emphasis, not the basic meaning.

A neutral version is:

• Ő a könyvtárban tanul.

Other orders are possible, for example:

• A könyvtárban tanul.
  Neutral, with no explicit pronoun.
• Tanul a könyvtárban.
  Also possible; a bit different in emphasis.
• Ő tanul a könyvtárban.
  Can emphasize that he/she is the one studying there.

Hungarian word order is strongly connected to what information is new, emphasized, or contrasted.

It is both.

Functionally, it is a place expression meaning in the library.
Grammatically, the noun is in a case form with the ending -ban.

Hungarian uses many such case endings instead of English-style prepositions. So rather than a separate word for in, the language often puts that meaning directly on the noun.

A rough pronunciation guide for English speakers is:

• Ő ≈ a long rounded vowel, something like French eu in deux but held longer
• a ≈ short aw
• könyvtárban ≈ kuhnyv-taar-ban
• tanul ≈ taw-nool

A few helpful notes:

• ő is a long vowel
• ny sounds like ny in canyon
• á is a long a
• stress in Hungarian usually falls on the first syllable of each word

So the stress pattern is roughly:

• Ő
• a KÖNYVtárban
• TA nul

Yes. Although -ban literally means in, English often translates this naturally as at the library when talking about location in a general sense.

So depending on context, Ő a könyvtárban tanul may be translated as:

• He/She is studying in the library.
• He/She is studying at the library.

The Hungarian form still uses the in-type location ending.

Yes, you could add what the person is learning or studying.

For example:

• Ő a könyvtárban magyarul tanul. = He/She is learning Hungarian in the library.

Here:

• magyarul = in Hungarian / Hungarian as a language adverbially, meaning learning Hungarian

If you want to say the person is simply doing schoolwork in the library, tanul alone is already enough.