Uter liber tibi gratior est, hic an ille?

Uter means which of two? It is used when there are exactly two choices, as here: hic (this one) and ille (that one).

By contrast, quis or qui would be used more generally for who?/what?/which? without the special idea of one out of two.

So:

• uter = which of the two?
• quis/qui = who?/what?/which? in a broader sense

Uter is in the nominative singular masculine because it goes with liber (book), which is also nominative singular masculine.

The sentence is essentially asking:

• Uter liber ... est?
• Which book ... is?

Since liber is the subject, uter agrees with it in gender, number, and case.

Gratior is the comparative form of gratus.

• gratus = pleasing, welcome, agreeable
• gratior = more pleasing, more welcome

So tibi gratior est means is more pleasing to you or more naturally in English, you like better.

Latin often uses pleasing to someone where English prefers someone likes something.

Because gratus and gratior take the dative of the person affected.

So:

• tibi = to you
• gratior est tibi = is more pleasing to you

Latin says something like Which book is more pleasing to you?
English often turns that into Which book do you like better?

If Latin used te, that would be the accusative, and that is not what this adjective normally requires.

Because the subject is singular: uter liber = which book.

Even though two books are being compared, the question asks about one book at a time: which book is more pleasing? Therefore the verb is singular:

• est = is

They are the two alternatives being compared:

• hic = this one
• ille = that one

They stand for this book and that book without repeating liber.

So the full sense is:

• Which book is more pleasing to you, this one or that one?

Latin often omits a noun when it is obvious from context.

Here an means or in a direct question that offers alternatives.

So:

• hic an ille? = this one or that one?

This is a very common Latin way to set up an either/or choice in a question.

Yes, Latin could in theory repeat the noun for clarity, but it usually does not need to when the meaning is obvious.

So instead of repeating book, Latin simply says:

• hic an ille

That is natural and elegant Latin.

Yes. The comparative adjective gratior implies comparison between the two options.

Latin does not need to say than the other explicitly here, because the choice hic an ille already makes the comparison clear.

So the sense is:

• Which of the two books is more pleasing to you?

Latin word order is more flexible than English because the endings show the grammar.

This sentence begins with uter liber to highlight the main question: which book. Then it gives tibi gratior est (is more pleasing to you), and finally the two options: hic an ille.

A more literal order in English would be:

• Which book to you more pleasing is, this one or that one?

That sounds unnatural in English, but it is normal in Latin.

Yes, but in this sentence this one and that one are usually better in English, because the noun book is not repeated after them.

So:

• hic = this one
• ille = that one

If the noun were expressed, then this book and that book would work.

Yes, they are demonstratives, and they often carry a sense of distance:

• hic = this, the one near me / the one here
• ille = that, the one over there / the one farther away

Sometimes the contrast is physical, sometimes just mental or discourse-based. In a basic sentence like this, the simplest translation is just this one or that one.

A very literal translation is:

• Which book is more pleasing to you, this one or that one?

A more natural English translation might be:

• Which book do you like better, this one or that one?

Both reflect the Latin well.