J'ai trois livres dans mon sac.

In French, when je comes before a verb that starts with a vowel sound (like ai), it usually contracts to j' to make pronunciation smoother.

• Je ai → J'ai
• Je habite → J'habite
• Je écoute → J'écoute

This is called elision. The apostrophe marks the missing vowel e in je. You must write J'ai, not Je ai.

Ai is the first-person singular (I) form of the verb avoir (to have) in the present tense:

• j'ai – I have
• tu as – you have
• il/elle a – he/she has
• nous avons – we have, etc.

French uses avoir to express possession, just like English uses to have. So J'ai trois livres literally means I have three books. You would not use être (to be) for possession.

In French, nouns are made plural when the quantity is more than one, even when there is a number in front. So you must add -s to the noun:

• un livre – one book
• deux livres – two books
• trois livres – three books

The number does not replace the plural ending; you still need livres with an s.

With a specific number (un, deux, trois, etc.), you normally do not use an article:

• J'ai trois livres. – I have three books.
• J'ai deux frères. – I have two brothers.

Compare:

• J'ai des livres. – I have (some) books.
• J'ai les livres. – I have the books (specific ones).

Once you put a number, you drop des or les unless you have a very special structure (which is rare for beginners).

French possessive adjectives agree with the thing possessed, not with the owner.

• sac is a masculine singular noun.
• The masculine singular form of my is mon.

So:

• mon sac – my bag (masculine singular)
• ma table – my table (feminine singular)
• mes livres – my books (plural, masculine or feminine)

You cannot say ma sac because sac is masculine, and you cannot say mes sac because sac is singular.

In J'ai trois livres dans mon sac, the word sac is masculine. You can tell from:

• the possessive adjective: mon sac (masculine) not ma sac
• the dictionary: French dictionaries list nouns with m or f (e.g. un sac (m.))

There is no reliable rule from the word’s ending; you generally need to learn the gender together with the noun: un sac, le sac.

Dans is the usual preposition meaning in / inside when you talk about physical location:

• dans mon sac – in my bag
• dans la boîte – in the box
• dans la voiture – in the car

You would not say en mon sac in modern French; that sounds archaic or poetic.

You could, however, say something slightly different:

• Il y a trois livres dans mon sac. – There are three books in my bag.

This changes the structure but keeps dans.

Yes, there is a nuance:

• J'ai trois livres dans mon sac.
  Focuses on what you have. It emphasizes your possession of the books.

• Il y a trois livres dans mon sac.
  Literally There are three books in my bag.
  Focuses more on what is located inside the bag, less on you as the owner.

In many everyday situations they can both be used, but the emphasis is slightly different.

Rough pronunciation (in English-like syllables):

• J'ai → like zhay (the j sounds like the s in measure)
• trois → roughly trwah (the oi like wah, final s is silent)
• livres → roughly leevr
  
  • li like lee
  • v as in English
  • final -es is not fully pronounced; the r is a French guttural r
• dans → roughly dahn (nasal an; final s is silent)
• mon → roughly mohn (nasal on)
• sac → like sak (clear k sound at the end)

Spoken smoothly: zhay trwah leevr dahn mohn sak.

Yes. Livre can mean:

1. book (the meaning used here)

   • un livre – a book
   • trois livres – three books

2. pound, as a unit of weight (particularly in some contexts, or historically)

   • une livre de tomates – a pound of tomatoes

In your sentence, context and the preposition dans mon sac clearly point to books, not weight.

The sentence follows typical French word order:

1. Subject – Je (contracted to J')
2. Verb – ai
3. Direct object – trois livres
4. Location (prepositional phrase) – dans mon sac

So the pattern is: Subject + Verb + Object + Place
This is very similar to English: I have three books in my bag.