Well, I'm not sure there are any rules. Consider the two sentences:
Liber est meus "The book is mine"
Est mihi liber "I own a/the book" (literally "There is to me a book")
The first emphasizes the identity of the owner of the book (whose book is it); it would be even more emphatic if one reversed the words: meus est liber: it's my book. The second emphasizes, as it were, the fact of possession (I own this book). Both use the same nouns and pronouns, and the same verb (esse to be), but in different ways and with different emphases. Essentially, the first answers the question "Whose book?" the second the question "What do you own?"
As to when you can omit the verb: pretty freely. So long as the sense is clear, the verb esse can often be omitted, and frequently will be where the writer is aiming for a short, pithy, epigrammatic sort of sentence.