Learn Hardcore French: Pendant ce circuit, il montre le café où un célèbre philosophe écrivait ses livres. - During this tour, he shows the café where a famous philosopher used to write his books.

pendant = during (used with a noun or time period)
ce = this (demonstrative adjective for a masculine singular noun)
circuit = here means something like an organized tour, route, or itinerary, not an electrical circuit.

So in this context, circuit is a bit of a false friend: it does not primarily mean an electrical circuit, but rather a tour with several stops along a route (for example, a guided city tour).

Could you leave out ce and just say pendant circuit?

No. In French, a singular countable noun almost always needs a determiner (article, demonstrative, etc.).

You need something like:

pendant ce circuit – during this tour
pendant le circuit – during the tour
pendant un circuit – during a tour

Pendant circuit by itself is ungrammatical.

Why is pendant ce circuit placed at the beginning of the sentence? Could it go somewhere else?

French often places time expressions at the beginning of the sentence, just like English:

Pendant ce circuit, il montre le café ...
= During this tour, he shows the café ...

You could also say:

Il montre le café ... pendant ce circuit.

Both are correct. Putting pendant ce circuit at the start just sets the time frame right away and sounds very natural in narrative or descriptive French. The meaning does not really change.

What exactly does il montre mean here? Could we add an object like il nous montre?

Il montre = he shows.

il = he (very likely the guide in this context)
montre = present tense of montrer (to show)

The direct object is le café:

Il montre le café. = He shows the café.

If you want to specify whom he is showing it to, you can add an indirect object pronoun:

Il nous montre le café. = He shows us the café.
Il vous montre le café. = He shows you (plural / formal) the café.

The original sentence is fine without that indirect object because the context (a guide talking to tourists) usually makes it obvious.

Why do we use où after le café and not que or dans lequel?

In this sentence, où is a relative pronoun that means where and refers to le café (a place):

le café où un célèbre philosophe écrivait ses livres
= the café where a famous philosopher wrote his books

Use où:

when the thing before it is a place (café, ville, pays, maison, etc.)
or a time expression (le jour, le moment, l’époque, etc.)

Using que would be wrong here:

✗ le café que un célèbre philosophe écrivait ses livres – ungrammatical

If you really want to avoid où, you could say:

le café dans lequel un célèbre philosophe écrivait ses livres
(literally: the café in which a famous philosopher wrote his books)

But où is shorter and much more natural.

Does où always mean where? Can it also mean when?

Où is mainly where, but it can also refer to time expressions and be translated as when in English.

For places:

le café où il écrivait = the café where he wrote

For times:

le jour où il est arrivé = the day when he arrived
l’époque où ils vivaient à Paris = the time when they lived in Paris

In your sentence, où clearly refers to le café, so it is where.

Why is écrivait (imperfect) used instead of a écrit (passé composé)?

Écrivait is in the imparfait, which often expresses:

habitual actions in the past
ongoing or repeated actions
background description

Here, écrivait ses livres suggests that the philosopher used to write his books there regularly, or that writing there was a usual activity over a period of time.

... où un célèbre philosophe écrivait ses livres.
= where a famous philosopher used to write his books / would write his books.

If you said a écrit, it would point more to a completed event (or set of events) rather than a repeated habit.

How would the meaning change if we said ... le café où un célèbre philosophe a écrit ses livres?

With a écrit (passé composé), the focus shifts to completed actions:

... le café où un célèbre philosophe a écrit ses livres.
= the café where a famous philosopher wrote his books (and finished them there)

This can sound more like:

he completed his books there, or
his writing of the books is seen as a block of finished events.

With écrivait, the idea is more:

he used to write there,
he spent time there writing, in a more descriptive, habitual way.

Both are grammatically correct; the choice depends on whether you want to highlight habit/background (écrivait) or completed acts (a écrit).

Why is it ses livres and not les livres? What does ses refer to?

Ses is a possessive adjective, meaning his or her (for plural nouns):

ses livres = his/her books

Here, ses refers to un célèbre philosophe:

ses livres = the philosopher’s books

If you said les livres:

écrivait les livres = wrote the books (some specific books that we already know about from the context)
it would drop the idea of ownership; we would not know they are the philosopher’s own books.

So ses livres makes it clear they are the books that belong to that philosopher.

Why do we use ses and not son or sa? How do I choose the right possessive form?

In French, possessive adjectives agree with the thing possessed, not with the person who owns it.

son – before a singular masculine noun
sa – before a singular feminine noun
ses – before plural nouns (masculine or feminine)

Here:

livres is plural → we must use ses:
- ses livres = his/her books

Some examples:

son livre – his/her book (one book, masculine noun)
sa voiture – his/her car (one car, feminine noun)
ses livres, ses voitures – his/her books, his/her cars (plural nouns)

Why is célèbre placed before philosophe? I thought French adjectives usually go after the noun.

The general rule is:

many adjectives go after the noun,
but some can go before or before and after, with subtle differences.

Célèbre is one of those adjectives that can appear before or after the noun:

un célèbre philosophe – a famous philosopher
un philosophe célèbre – a famous philosopher

Both are correct and very common. Any difference is very subtle:

un célèbre philosophe can sometimes sound a bit more like a well‑known, celebrated philosopher (slight emphasis on renown).
un philosophe célèbre may feel slightly more descriptive or neutral.

In practice, you will hear both; the choice here is mostly stylistic.

Are there any pronunciation tips for this sentence?

Yes, a few key points:

Pendant ce circuit
- pendant → final t is usually silent: pɑ̃-dɑ̃
- ce → like sə
- circuit → final t also silent: roughly seer-kɥi
il montre
- il → often pronounced very lightly, almost eel
- montre → final -e is silent; mon-tr (nasal on sound)
le café
- café → stress on the second syllable: ka-FÉ
où
- has the ou sound like in vous, not like in eu (as in peu).
écrivait
- é at the start and ait at the end both sound like é:
  é-kri-vé
ses livres
- ses → sounds like ces and sais (sé). Context tells you which it is.
- livres → final s is silent: leevr.

Try saying the sentence with smooth rhythm and no English r sound; use the French uvular r in circuit, écrivait, livres.