Learn Hardcore French: C'est Paul qui comprend le mieux ce livre. - It is Paul who understands this book best.

Using C'est Paul qui… is a very common French structure called a cleft sentence.
It serves to emphasize the element after C'est, here Paul.

Paul comprend le mieux ce livre = Paul understands this book best.
C'est Paul qui comprend le mieux ce livre = It is Paul (and not someone else) who understands this book best.

The second version puts strong focus on Paul as the one who has this quality.

What exactly is the role of qui in this sentence?

Here qui is a relative pronoun that refers back to Paul and introduces the clause qui comprend le mieux ce livre.

It stands for Paul inside the second part of the sentence.
It is the subject of the verb comprend.
It cannot be omitted in French; you cannot say C'est Paul comprend le mieux ce livre.

So the structure is: C'est Paul [subject] qui comprend le mieux ce livre [relative clause about Paul].

Why is it C'est Paul and not Il est Paul?

In French, when you identify someone with a name, you normally use C'est, not Il est.

C'est Paul = It is Paul / That’s Paul.
Il est Paul is not natural French in this context.

Il est is used before adjectives, professions without an article, and some short descriptions, for example:

Il est intelligent.
Il est professeur.

But with a proper name, you use C'est Paul.

What does le mieux mean here, and why is there a le?

Mieux is the comparative form of bien (well):

bien → well
mieux → better

Le mieux is the superlative form:

le mieux → the best (in the sense of best as an adverb: understands the best).

The le is needed to form the superlative adverb:

mieux = better
le mieux = the best.

So comprend le mieux means understands best.

Why is it le mieux and not le meilleur?

Mieux and meilleur both translate to better/best, but:

Mieux is an adverb and modifies verbs (how something is done).
Meilleur is an adjective and modifies nouns (what something is like).

In comprend le mieux, we are modifying the verb comprend (understands), so we must use the adverb mieux:

Il comprend le mieux. → He understands best.
C’est le meilleur livre. → It is the best book.

So le mieux is correct here because it describes how he understands.

Does le in le mieux agree with anything? Why masculine and not feminine?

In le mieux, the le is not really agreeing with a noun the way a normal article does.
It is part of a fixed structure for the superlative of adverbs:

le mieux (from bien)
le plus vite (the fastest)
le moins bien (the least well)

This le is often called a neutral or invariable article.
It does not change to feminine or plural; you do not say la mieux to match personne or anything like that in this kind of structure.

Why is it comprend and not comprends or another form?

Comprend is the third person singular of comprendre in the present tense:

je comprends
tu comprends
il / elle / on comprend
nous comprenons
vous comprenez
ils / elles comprennent

Because the subject (inside the relative clause) is qui, which refers to Paul (he), the verb must be il comprend → qui comprend.

So C’est Paul qui comprend is grammatically C’est Paul. Il comprend le mieux ce livre.

What is the nuance of using C'est Paul qui… instead of C'est lui qui…?

Both are possible, but there is a slight nuance:

C’est Paul qui comprend le mieux ce livre.
– Names him explicitly, maybe to tell someone which person it is.
C’est lui qui comprend le mieux ce livre.
– Refers to him as he, often when the person is already known or visible.

In many contexts they are interchangeable, but C’est Paul… is more explicit and identifying, while C’est lui… assumes the listener already knows which person is being singled out.

Can the word order be changed, for example C'est Paul qui ce livre comprend le mieux?

No, that word order is incorrect in standard French.

The normal order inside the clause after qui is the usual French Subject – Verb – Object:

qui (subject)
comprend (verb)
le mieux (adverb)
ce livre (object)

So you should keep: qui comprend le mieux ce livre, not qui ce livre comprend le mieux.

Could you say Paul est celui qui comprend le mieux ce livre instead? What is the difference?

Yes, Paul est celui qui comprend le mieux ce livre is correct and means almost the same thing.

Nuance:

C’est Paul qui comprend le mieux ce livre.
– Very common, neutral, and slightly more conversational.
Paul est celui qui comprend le mieux ce livre.
– A bit more formal or explicit; it literally says Paul is the one who understands this book best.

Both clearly highlight Paul, but the C’est Paul qui… structure is more frequent in everyday speech.

Is C'est Paul qui comprend le mieux ce livre formal or informal?

This structure is neutral; it works both in spoken and written French and in most registers.

Perfectly normal in casual conversation.
Also completely acceptable in essays, reports, and more formal speech.

It is a very natural and common way to emphasize a particular element.

How would you make this sentence negative, for example It is not Paul who understands this book best?

To make it negative, you put ne … pas around est:

Ce n’est pas Paul qui comprend le mieux ce livre.

The rest of the sentence stays the same.
The pattern is: Ce n’est pas X qui…

How would you ask Who understands this book best? using a similar structure?

You can use a cleft question with qui:

Qui est-ce qui comprend le mieux ce livre ?

Here:

Qui = who (the thing you are asking about)
est-ce qui introduces the clause
comprend le mieux ce livre is the rest of the sentence

This structure is very common in spoken French.