Learn Hardcore Portuguese (Brazil): Vários alunos gostam de estudar na biblioteca. - Several students like to study in the library.

So gostam de estudar is literally they like of studying, but in English we translate it simply as they like to study or they like studying. The de is required in Portuguese even though English doesn’t have a preposition here.

Why is it estudar and not estudando after gostar de?

After gostar de, the natural form in Portuguese is the infinitive (estudar), not the gerund (estudando).

Vários alunos gostam de estudar na biblioteca.
= Several students like to study in the library / like studying in the library.

Using estudando here would sound wrong:
✗ Vários alunos gostam de estudando na biblioteca.

So, with gostar de + verb, keep the verb in the infinitive:

gostar de viajar – to like traveling
gostar de ler – to like reading
gostar de comer – to like eating

Does Vários alunos gostam de estudar na biblioteca mean they are studying there right now?

No. It talks about a general preference or habit, not about what is happening at this moment.

Vários alunos gostam de estudar na biblioteca.
= They generally like / prefer studying in the library.

If you want to say they are studying there right now, you would use a progressive form:

Vários alunos estão estudando na biblioteca.
= Several students are studying in the library (right now).

Why is the verb gostam (plural) and not gosta (singular)?

The verb agrees with the subject vários alunos, which is plural.

vários alunos → eles (they) → gostam
um aluno → ele (he) → gosta

Compare:

Vários alunos gostam de estudar na biblioteca.
Several students like to study in the library.
Um aluno gosta de estudar na biblioteca.
One student likes to study in the library.

What exactly is vários here? How is it different from muitos?

Vários is a quantifier meaning several or a number of. It suggests more than a few, but not necessarily a huge amount.

vários alunos – several students, a number of students
muitos alunos – many students, a lot of students

Both are plural, both go before the noun, and both agree in gender and number:

vários alunos (masculine plural)
várias alunas (feminine plural)
muitos alunos
muitas alunas

Nuance:

vários: neutral, “several / some (more than just a few)”
muitos: emphasizes quantity, “many / a lot of”

Could I say Os vários alunos gostam de estudar na biblioteca? Do I need an article with vários alunos?

Normally, you do not use an article here; vários alunos by itself is most natural:

Vários alunos gostam de estudar na biblioteca. ✅

You could say Os vários alunos..., but it sounds more marked and specific, like:

Os vários alunos que conheço gostam de estudar na biblioteca.
The several students that I know like to study in the library.

So:

vários alunos – general, “several students”
os vários alunos – “those several students” (already known or identified in context)

What is na in na biblioteca? Why not just em biblioteca?

Na is a contraction:

na = em + a

So:

em = in / on / at
a = the (feminine singular article)
em + a biblioteca → na biblioteca

Portuguese normally uses an article with countable, specific nouns like biblioteca:

na biblioteca – in the library
no banco – in the bank / at the bank (no = em + o)
na escola – at school (na = em + a)

Saying em biblioteca without article would sound unnatural in this context. English sometimes drops the (e.g., “at school”), but Portuguese usually keeps the article.

When do I use no and when do I use na?

No and na are both em + definite article, but:

no = em + o (masculine singular)
na = em + a (feminine singular)
nos = em + os (masculine plural)
nas = em + as (feminine plural)

Examples:

no banco – at the bank (masc. sing.)
na biblioteca – in the library (fem. sing.)
nos bancos – in the banks (masc. pl.)
nas bibliotecas – in the libraries (fem. pl.)

How is vários pronounced, and what does the accent mark in vários do?

Vários is pronounced approximately like: VAH-ree-oos (Brazilian Portuguese).

Phonetic (Brazilian): [ˈva.ɾi.us]

The acute accent (á) shows:

The syllable vá is stressed.
The a is an open vowel sound (like the “a” in “father”).

So:

vários – stress on the first syllable: VÁ-rios
Without the accent, the stress would move (and it would be a different word or incorrect spelling).

What is the difference between alunos and estudantes?

Both can translate as students, but there is a nuance:

aluno / aluna – a student enrolled in a school, course, or with a specific teacher. More tied to a formal learning context.
estudante – student in a broader sense: a person who studies, often used for older students (university) but also general.

In your sentence:

Vários alunos gostam de estudar na biblioteca.

This suggests students of some school/class/course like to study in the library. You could also say:

Vários estudantes gostam de estudar na biblioteca.

This is also correct and very natural; it just sounds a bit more general.

Does na biblioteca describe gostam or estudar? Do they like the library, or do they like studying there?

Here, na biblioteca is understood as describing estudar (where they study), not directly gostam.

They like studying in the library.

If you wanted to say they like the library itself, you’d usually make the library the object of gostar de:

Vários alunos gostam da biblioteca.
Several students like the library.

So:

gostar de estudar na biblioteca → like the activity of studying in that place
gostar da biblioteca → like that place itself

Could I say Vários alunos gostavam de estudar na biblioteca? What changes?

Yes, that is correct. You are just changing the tense.

gostam – present tense:
Vários alunos gostam de estudar na biblioteca.
Several students (currently / generally) like to study in the library.
gostavam – imperfect past:
Vários alunos gostavam de estudar na biblioteca.
Several students used to like studying in the library / would like studying in the library (habit in the past).

So the structure is the same; only the time reference changes.