Learn Hardcore Czech: V našem městě je nová knihovna, kde si můžu půjčit hodně knih. - In our city there is a new library where I can borrow many books.

nová knihovna literally just means new library, and context tells you whether it’s a new library or the new library.
So V našem městě je nová knihovna is understood as There is a new library in our town without needing an article word.

Why is it v našem městě and not something like v naše město?

Because the preposition v (“in”) needs the locative case when it means location.

The base forms are naše město = our town (nominative case).
After v (in) for location, město changes to městě (locative singular).
The adjective naše must agree in gender, number, and case with město, so it becomes našem (neuter, singular, locative).

So:

naše město (nominative) → v našem městě (locative after v = “in our town”).

How do našem and městě agree with each other?

They match in three things:

Gender: město is neuter, so náš → naše → našem is in the neuter form.
Number: město is singular, so the adjective is singular too.
Case: after v (location) we need locative, so both are locative:
- naše → našem (neuter, singular, locative)
- město → městě (neuter, singular, locative)

Could I also say Nová knihovna je v našem městě? Is there a difference?

Yes, that’s a correct sentence too.

V našem městě je nová knihovna = more like In our town (there) is a new library. It emphasizes the place first.
Nová knihovna je v našem městě = The new library is in our town. It starts from the library and then says where it is.

Both are grammatical; the choice depends on what you want to highlight first (location vs. the library).

Why is there a comma before kde?

kde introduces a relative clause (a dependent clause that describes the library), so Czech normally uses a comma:

Main clause: V našem městě je nová knihovna
Relative clause: kde si můžu půjčit hodně knih

In English, that’s like:

…there is a new library, where I can borrow a lot of books.

Czech uses the comma to mark off this additional descriptive clause, just as English often does with where, which, who, etc.

Does kde here only mean “where”, or is it more like “where/at which”?

Literally kde means where, but in a relative clause like this you can understand it as:

where
in which / at which

So nová knihovna, kde si můžu půjčit hodně knih =
a new library where I can borrow many books / in which I can borrow many books.

What is the function of si in kde si můžu půjčit hodně knih?

si is a clitic reflexive pronoun in the dative. With the verb půjčit it’s essential for the meaning to borrow:

půjčit někomu něco = to lend someone something
- e.g. Půjčím ti knihu. = I’ll lend you a book.
půjčit si něco = to borrow something (for oneself)
- e.g. Půjčím si knihu. = I’ll borrow a book.

So in your sentence, půjčit si means to borrow (for myself). Without si, půjčit would more likely be understood as to lend.

Why is si placed after kde and before můžu? Can I say kde můžu si půjčit?

You cannot say kde můžu si půjčit. In Czech, short unstressed words like si (called clitics) usually go in second position in the clause:

Clause starts: kde
Second position: si
Then: můžu půjčit …

So the normal orders are:

kde si můžu půjčit hodně knih (most natural)
kde můžu hodně knih půjčit si is possible but sounds odd and unnatural.

The key rule: si wants to stand very early in the clause, typically right after the first stressed element (here: kde).

What’s the difference between můžu and mohu?

Both mean “I can / I am able to / I may” (1st person singular of moci).

můžu – more colloquial, everyday spoken Czech.
mohu – more formal, careful, or written style.

In this sentence, kde si můžu půjčit hodně knih is perfectly natural conversational Czech. In a very formal text you might see kde si mohu půjčit hodně knih.

Why is it půjčit and not půjčovat?

This is about aspect:

půjčit (si) – perfective: a single, completed act of borrowing.
půjčovat (si) – imperfective: repeated / ongoing borrowing, or focusing on the process.

In můžu si půjčit hodně knih, the idea is I am able to (go there and) borrow (some books) – one action each time. If you said:

můžu si půjčovat hodně knih – it would suggest I can keep borrowing lots of books (again and again), emphasizing repeated activity.

Both are possible, but půjčit is the more neutral choice here.

Why is it hodně knih and not hodně knihy?

Because hodně (“a lot of, many, much”) requires the genitive case:

kniha (nominative singular)
knihy (nominative plural / genitive singular)
knih (genitive plural)

After hodně, you need genitive plural, so:

hodně knih = a lot of books

Other examples:

hodně lidí = a lot of people (genitive plural of člověk → lidé / lidí)
hodně času = a lot of time (genitive singular of čas).

What’s the difference between knihovna and knihkupectví? Both are related to books, right?

Yes, both are connected to books, but:

knihovna = library (a place where you borrow books, usually for free)
knihkupectví = bookstore / bookshop (a place where you buy books)

So your sentence is specifically about a library, not a shop.

Why is the subject “I” not written? Where is já?

Czech is a pro‑drop language: the subject pronoun is often omitted because the verb ending already shows the person.

můžu is 1st person singular (“I can”).
Adding já is only needed for emphasis or contrast, e.g.
- Já si můžu půjčit hodně knih, ale ty ne. = I can borrow many books, but you can’t.

In a neutral sentence, just můžu is enough, so já is left out.