About the only three vocalic sounds for the standard Arabic pronunciation, you can check any reliable source for Arabic language. I have here the Gramática árabe, by Corriente (pp. 10-39), but the same information is ofered at Wikipedia:http://en.wikipedia.org/wiki/Arabic_phonology#Vowels
And about the apparition of the non-standard vowels [e] and [o] in some Arabic dialects, as Moroccan:http://en.wikipedia.org/wiki/Moroccan_Arabic#Vowels
As তন্ময় ভ said, when someone ask for *the* Arabic pronunciation (as it was the case of the OP), the pronunciation asked is the standard classical pronunciation (conserved in the Quran); specially when asking for "the correct pronunciation". Other dialectal pronunciations are possible, but in those cases it is necessary to put the gentilice before "Arabic": Moroccan Arabic, Lebanese Arabic, Egyptian Arabic, Syrian Arabic...
Leyla/Leila represents the Persian pronunciation ['lejla] (LAY-la, where LAY rhymes with "day" or "may"), adaptation from the Arabic Laila
. This form of the name (and the Arabic original form) are known since the Middle Ages, known by Persian tales.
The Arabic pronunciation ['lajla] (LY-la, where LY rhymes with "my" o "by") was rendered mainly as Layla/Laila. In English, however, and due to the pronunciation of the group AY like [ej] (day, may), Layla/Laila is also usually read ['lejla]; that is why it is not unusual find that both spellings (with A and with E) are variants. In fact, they are cognates but not variants in the same origin language.
You can check Diccionario de nombres propios
, by Roberto
Faure, and manuals of European medieval litterature about the transmission of the Leila
and Majnun story and One Thousand and One Nights tales in Europe