Learn Hardcore Russian: У меня есть четыре книги.

Literally, the phrase breaks down as at me there is four books (“у” + genitive “меня” means “at me,” “есть” is “there is/are,” and “четыре книги” is “four books”). Idiomatically in English we say I have four books.

Why do we use У меня есть to express “to have,” instead of a verb like иметь?

Russian doesn’t use a direct equivalent of the English verb “to have” in everyday speech. Instead, possession is shown with the construction У + [genitive of the possessor] + есть. The verb иметь exists but sounds bookish or bureaucratic; native speakers virtually always say У меня есть rather than Я имею.

Can the verb есть be omitted? What happens if I just say У меня четыре книги?

Yes. In colloquial Russian you normally drop есть, so У меня четыре книги is perfectly natural and means exactly the same thing: I have four books. Including есть can add a slight emphasis on existence (e.g. contrasting with not having something).

Why is книги not in the nominative plural but in the genitive singular form?

Russian counting rules say that after the numerals 2, 3, and 4 the noun takes the genitive singular. Even though nominative plural of “книга” happens to look the same (книги), you recognize it as genitive singular here because of the numeral четыре.

What case is четыре in, and why isn’t it четырёх like in нет четырёх книг?

With есть the counted item is in the nominative, so the numeral stays in its nominative form четыре. In contrast, when you negate possession with нет, the whole phrase shifts to genitive, giving нет четырёх книг (genitive of четыре книги).

How would I ask “Do you have four books?” in Russian?

Switch the possessor to second person and use a question mark or rising intonation:
У тебя есть четыре книги?
Literally at you there is four books?, i.e. Do you have four books?

How do I say “I don’t have four books”?

Replace есть with нет and put both numeral and noun into the genitive:
У меня нет четырёх книг.
Literally at me there is no four books, i.e. I don’t have four books.

Is word order flexible in У меня есть четыре книги?

The normal order is У меня есть + [what you have]. You can occasionally stress a different part by moving it, e.g. Четыре книги у меня есть, but that sounds more emphatic or poetic rather than neutral.