I would like to know the meaning of the Hadeeth where the Prophet, sallallaahu ‘alayhi wa sallam, said that the Quran was revealed to be recited in seven different letters (ways).
Is this Hadeeth Saheeh (authentic)? Does this mean that the language of the Quraysh is the language of the Quran? If not, is the Bedouin language the language of the Quran?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu 'alyhi wa sallam, is His slave and Messenger.

The Hadeeth in question was narrated in the two Saheehs (collections of authentic narrations) on the authority of ‘Umar ibn Al-Khattaab . 

In the Hadeeth ‘Umar said that he heard Hishaam ibn Hakeem reciting Soorah Al-Furqaan in a way that was different to ‘Umar's recitation, which he had learnt from the Prophet, sallallaahu ‘alayhi wa sallam. Therefore ‘Umar was angry and seized him and dragged him to the Prophet, sallallaahu ‘alayhi wa sallam, and told him what had happened. The Prophet, sallallaahu ‘alayhi wa sallam, ordered Hishaam to recite the chapter and so he recited it the same way that ‘Umar had heard him reciting. Then the Prophet, sallallaahu ‘alayhi wa sallam, said that it was a correct recitation. He also ordered ‘Umar to recite and the Prophet, sallallaahu ‘alayhi wa sallam, judged his recitation as correct too. He then told them that the Quran had been revealed to be recited in seven different ways.  [Al-Bukhari and Muslim]

It was also narrated on the authority of Ibn ‘Abbaas both, that the Prophet, sallallaahu ‘alayhi wa sallam, said that Jibreel (Gabriel ) recited the Quran to him in one way. Then he, sallallaahu ‘alayhi wa sallam, requested him to recite it in another way. The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, continued asking him for more until Jibreel recited it in seven different ways. [Al-Bukhari and Muslim]

As for the status of this Hadeeth, it is clear that it is highly authentic, but scholars differ greatly on what is meant by the "seven different letters (ways)."

One of these opinions is that of Abu ‘Ubayd Al-Qaasim ibn Sallaam whose opinion was adopted by Ibn ‘Atiyyah who said that the seven letters (ways) refers to the different dialects used among Arabs at the time of the revelation, such as Quraysh, Huthayl, Hawaazin, Al-Yaman etc., and their different ways of pronunciation. He also argued that the meaning could not be that each letter could be recited in seven different ways.

To strengthen this opinion, its advocates argued that:

•          ‘Uthmaan ordered Mus-hafs (copies of the Quran) to be written according to the dialect of Quraysh if there was any argument about which dialect a verse should be written in, since, as he said, most of the Quran was revealed in the language of Quraysh.

•          Some companions were doubtful about the exact meaning of some Quranic words until they heard these words spoken in the dialects of other Arab tribes.  For example, this happened to Ibn ‘Abbaas regarding the meaning of the word Faatir (Creator) and the meaning of the verse in which Allaah the Almighty Says (what means): {Our Lord, decide between us and our people in truth, and You are the best of those who give decision.}[Quran 7:89]  

Again, the same thing happened to ‘Umar concerning the meaning of the verse in which Allaah the Almighty Says (what means): {Or that He would not seize them gradually [in a state of dread]? But indeed, your Lord is Kind and Merciful.}[Quran 16:47]  He understood its meaning after an Arab Bedouin explained it to him, referring to some lines of poetry.

All these examples show that the Quran was revealed to be recited in different Arab dialects besides the dialect of the Quraysh. However, most of it was revealed in the language of Quraysh.

In fact, Al-Qaadhi Abu At-Tayyib explained that there are words in the Quran that differ from the language of Quraysh, as Allaah the Almighty Says (what means): {Indeed, We have made it an Arabic Quran that you might understand.}[Quran 43:3] Thus it is not a 'Qurayshi' Quran.

It is also incorrect to claim that 'Arabic' in this verse means that the Quran was revealed in the dialect of the Quraysh or the language of any other Arab tribe, because the word 'Arabic' equally applies to all dialects of the Arab tribes.

This explanation provides the answer to the rest of your question.

And Allaah Knows best.