So, an has two meanings that (I believe) you've learned. Yes, it can act as the definite article. However, it's also the question particle in some tenses. The difference is whether it comes before a noun or a verb. So here, an is marking a question. Bhfuil the form of bí that goes after the question particle in the non-habitual present tense.
Also, a more literal translation is "Is there a book at you" (note, ag means "at", not "with").
Thanks for adding the example with the plural form. I'm finding the difference quite subtle. She is rolling the r in the singular but there is a slight trill (I think is the way to explain it) in the plural. This makes it quite similar to my ear. I will need to pay close attention to this to get it right.
Standard English doesn't differentiate between "singular "you" (talking to one person) and plural "you" (talking to multiple people). In many dialects of spoken English, though, there is a distinction made, such as "you all" for plural "you" in Southern US English, or "ye" in some parts of Ireland and Britain.
This distinction doesn't always make it into written English (even people who make the distinction in spoken English might not make it when writing), but it is useful when translating from Irish, where there is a distinction between tú (singular "you") and sibh (plural "you"). In this exercise agaibh tells us that the question is addressed to a group, so "do ye have" is accepted as an answer.
I am getting a little confused about the usage of "an bhfuil". Here it translates to "Do you have a book?" Another sentence "An bhfuil torthaí ón gcailín?" translates to "Does the girl want fruits?" Is it interchangeable between "want" and "have"? Or does some part of the context determine the meaning?
Irish doesn't have a verb that means "have", or a verb that means "want". Instead the phrasal verb tá ... ag is used to say "have" and tá ... ó is used to say "want".
Tá X ag Y - "Y has X"
Tá X ó Y - "Y wants X"
An bhfuil is the interrogative form of Tá, so An bhfuil X ag Y? means "Does Y have X?" and An bhfuil X ó Y? means "Does Y want X?".
The preposition used modifies the meaning of the verb.