Директор is one of the top guys, the leader, only answering to the owners of the company (that is, if the so-called Russian "director" is not THE owner). In a smaller company директор is the "CEO", the boss of almost everything, minus the stuff done by the accountant in charge (more precisely, it is «генера́льный дире́ктор»). It is understood, though, that most moderately-sized companies will have more than one manager, so it does not seem quite right to accept that.
Admittedly, there is no one-to-one correspondence. In layman's terms, the CEO is probably the closest match your are going to get, while a "manager" is needlessly vague.
The Nominative plural is директора́, with a stress on the last syllable. It is always on the last syllable with masculine nouns.
The regular pattern usually has "typical" masculine nouns (e.g., стол, компьютер, ботинок, нож) end in -ы(-и) in plural: столы, компьютеры, ботинки, ножи. However, quite a number of masculine nouns have a stressed -а(-я) as their ending. Some of these are historical, coming from Dual number (глаз → глаза́, рог → рога́). Some of these are newly formed, first coming into colloquial use and then being accepted as a norm. Consider it an aspiring new regular pattern for some nouns.
The most important nouns of this kind for a beginner are:
- дом → дома́ (house)
- го́род → города́ (city)
- до́ктор → доктора́ (doctor)
- дире́ктор → директора́ (CEO, director)
- профе́ссор → профессора́ (professor)
- ма́стер → мастера́ (master, as in "skilled professional")
- по́езд → поезда́ (train)
- учи́тель → учителя́ (учи́тели is only for "spiritual" teachers)
- глаз → глаза́ (eye)
- цвет → цвета́ (colour)
- ве́чер → вечера́ (evening)
- го́лос → голоса́ (voice, vote)
- лес → леса́ (forest)
- па́спорт → паспорта́ (passport)
- про́пуск → пропуска́ (ID)
- но́мер → номера́ (ordinal number, hotel room)
- счёт → счета́ (account, bill)
- а́дрес → адреса́ (address)
These are truly irregular, some not even ending-stressed:
- друг → друзья́ (friend)
- сын → сыновья́ (son)
- брат → бра́тья (brother)
- стул → сту́лья (chair)
This mostly covers most popular 1000 words in Russian, and actually a bit beyond that.
"Вашего" is the singular genitive form of "ваш" and requires the noun that follows to be the same. However, with the last syllable stressed in the recording ("директоРА" is plural nominative), the sentence is grammatically incorrect. Singular genitive (and accusative) is "диРЕКтора."
To translate possessive situations into Russian, first convert sentences with 's into sentences with "of". Then the word(s) after "of" will be in genitive case and the rest of the sentence should maintain its word order.
So: What is your director's name? =>
What is the name of your director? =>
Как зовут вашего директора?