Every Prophet and Messenger sent by God; may He be exalted was given a miracle relevant to the people of that time. Similarly, Qura’n is a miracle that was given to the last and final messenger of All-Mighty God Allah, Prophet Muhammad, may peace and blessings be upon him. 

The only difference between the other prophets and Prophet Muhammad (peace be upon Him) is that the other prophets were meant only for a group of people and for a particular period of time, but Prophet Muhammad was sent for all of mankind and the Quran was meant to be followed by every human being regardless of his nationality, race and ethnicity as Allah says in Surah Saba Chapter 34 Verse 28:

And We have not sent you except comprehensively to all of mankind as a bringer of good tidings and a warner. But most of the people do not know.

It was narrated from Abu Hurayrah that the Messenger of Allah (blessings and peace of Allah be upon him) said: "There is not a single Prophet who was not given signs so that the people would believe in him because of them. What I have been given is a Revelation that Allah has revealed to me, and I hope that I will be the one with the most followers on the Day of Resurrection."

Narrated by al-Bukhaari (1044) and Muslim (152).
 
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: There are several possible interpretations of the hadeeth:
 
It was said that what is meant is that the miracles of the Prophets have come to an end with the end of their time, and no one witnessed them except those who were present, but the miracle of the Qur’an is ongoing and will continue until the Day of Resurrection. It is extraordinary in its style and eloquence, and in what it foretold of future events of the unseen, so that no era passed except something appeared that the Qur’an said would come to pass, which proves the truthfulness of its claim. This is the strongest of the interpretations of the hadeeth. 

The answer to the question, how can the Qur'an be a miracle for the unlettered non-Arabs who formed the majority of inhabitants of the world at that time? is as follows:

It is known that the Prophet (blessings and peace of Allah be upon him) was sent among the Arabs, who were distinguished by their great eloquence and beautiful style in speech, so Allah, may He be exalted, made the miracle or sign of the Prophet (blessings and peace of Allah be upon him) of the same nature of that by which his people were distinguished, so that the miracle and challenge would be more effective, just as the sign of Moosa (peace be upon him) was his staff or stick, with which he overcame the deceit of the sorcerers, and the sign of ‘Eesa (peace be upon him) was the healing of those who had been born blind and of the lepers, because knowledge of medicine was widespread at his time. 

With regard to the non-Arabs, in the past and at present, and how the Qur’an may be a miracle for them, and whether they were included in its challenge, not all Arabs have great knowledge of the Arabic language and its style and eloquence, just as not all non-Arabs are ignorant of the Arabic language. Hence it is known that the miracle and the challenge is addressed to all those who know Arabic, whether they learned it from birth, like the Arabs, or they learned it later on, like the non-Arabs. Thus, it is clear that the case of the non-Arabs is like that of Arabs who have no knowledge of their language. 

The Arabic language has a very complex system of grammar and word formation. One statement can mean many things, and this is only possible in the Arabic language. Arabic is syntactical which means it cannot be compared to any other languages in terms of grammatical rules.

The shortest wording without any loss in the meaning is only in the Arabic language. It is also the most advanced language amongst world languages in terms of eloquence, literature and fluency. Thus, no other language can express the meanings in the world.