"Her husband reads a book."
Translation:Ο άντρας της διαβάζει ένα βιβλίο.
Is there a rule to distinguish the two different meanings, if we found the phrase "Ο άντρας της διαβάζει ένα βιβλίο"?
Your sentence means "Her husband is reading a book."
To give the meaning "The husband is reading her a book", you're supposed to write Ο άντρας τής διαβάζει ένα βιβλίο, with the τής accented to show that it's a personal pronoun (to her) rather than a possessive.
The pronunciation of the individual words is the same, but the sentence intonation is different.
Clitics generally have no word stress of their own and are pronounce together with the following or preceding word, so "Her husband is reading a book" sounds a bit like οάντραςτης διαβάζει έναβιβλίο, while "The husband is reading her a book sounds a bit like οάντρας τηςδιαβάζει έναβιβλίο -- the της gets pronounced together with the verb rather than together with the noun.
Because of noun genders.
Each Greek noun is either masculine, feminine, or neuter, and you have to use the correct form of various words such as adjectives or articles together with a noun depending on the gender of the noun.
βιβλίο is a neuter noun, so ένα is acceptable (in fact, the only possible form of the indefinite article) and μία is not.
If he had been reading a newspaper (εφημερίδα, which is feminine), then μία would have been acceptable (or rather required) and ένα would have been wrong.
What is the difference of meaning "Ο αντρας της" and "Ο σύζυγος της"
In practice, not much.
ο άντρας της is "her husband"
ο σύζυγός γης is "her [male] spouse"
Since we call a male spouse a husband in English, the two are synonyms here.
Similarly with η γυναίκα του "his wife" versus η σύζυγός του "his [female] spouse".