As far as the conquests, the thing to realize is that there are absolutely enormous wars going on between Heraclius's Byzantium and Khosrau's Persia. They were weakening each other and discrediting each other: the Persians took Jerusalem, including the presumed True Cross; then, they suffered their own misadventures.

Meanwhile, infighting between the church leaders at Antioch, Alexandria, Constantinople, and Rome has everyone split up into teams, mostly interested in regional pride but ostensibly arguing over the composition of the Godhead, the nature of the Incarnation, the status of Jesus's mom, &c. &c. &c. The main things w/r/t Muhammad is that these guys are all denouncing and discrediting one another and that the version most popular near and within Arabia isn't the one that became orthodox in the west but mono- or miaphysitism (really complicated, but more or less the idea that Jesus was essentially much more God than man). The Qur‘an is stringently monotheist and reacts starkly against the polytheism seemingly creeping into the Trinity; it still threads the needle by making Jesus a prophet in the Biblical mode and by making a huge deal over Mary and still giving her a virgin birth.

But as to context as you read the Qur‘an itself, what's most important is having a guide as to the events within Muhammad's own life, since certain surahs are reactions to ongoing events as he struggles against the clans running Mecca's polytheistic tourist trap. I've picked it up piecemeal, but I'm sure some of the other editors here can give you some sources.

As you can see, you opened an historical can of worms with a simple question. But here's a quick, down-and-dirty primer for you.

The majority of Arab society at the time was divided in many ways; religiously, ethically, and even the vast urban/nomadic divide. but they all shared an idealized mode of behavior; be generous to your guests, be kind to your tribe-mates, be unforgiving to those who have wronged you, spirits and temptation abound, etc. Into that mix, you add the city of Mecca - like many other cities in and near the Red Sea, it was a transit point for the trade from Asia, Africa, and the Mediterranean but wanted to stand out (because more money is better than less money). As such, shrines and idols were erected in order to draw more merchants to Mecca than to other, competing cities. It worked and the tribe in charge of Mecca - the Qurayshi tribe - became incredibly rich. As they became more urban and more wealthy (well, some of them became more wealthy), they began to distance themselves from their traditional tribal ethics (mentioned above) to the point where the poor were often treated rather harshly and orphans had it the worst.

That's where Muhammad entered the scene; his father had died when he was unborn and his mother died when he was 6, leaving him technically an orphan. But his grandfather watched over him, taught him the traditional values of honesty, respect, and compassion even as he found Muhammad a job with a merchant company. As Muhammad grew up, he manifested a lot of the traditional values he had been instructed in as a child; charity, humility, generosity, respect, piety, etc. but was troubled by Mecca's urban life - hence why he meditated in the caves outside the city. It was in those caves that he is believed to have received the revelations which lead to the Qur'an.

Keep in mind also that when the Qur'an was being compiled, there were a lot of religious influences upon Arab/Bedouin society - Judaism, Christianity, Zoroastrianism, and the older Animist beliefs all had a part in forming the Qur'an's perspective. Also keep in mind that geography plays a huge part in the shape and tone of the Qur'an as well; At the heart of the document is life in the 7th century in one of the harshest places on earth. Finally, keep in mind that though Muhammad and Jesus are often compared to each other, they are completely different types of religious figures (one is not better than the other, just different). For example, Muhammad was not only a religious leader, but he was a community leader, a judge, and a military leader because his people needed him to be those things.

I hope you enjoy your exploration into the Qur'an; it is a brave, bold step you are taking and one which will doubtless be eye-opening.

Just for clarification, are you asking what was going on in the rest of the world (i.e. Mediterranean, Europe, Asia), or are you asking for what was going on in what is now considered the Arab world (Middle East, North Africa, Central Asia), or are you asking for what was going on in what was then considered the Arab World (Arabian peninsula only)?

you should definitely read Reza Aslan's No God but God, fantastic book about the time period. It has been a few years since i have read it so I cant summarize it well but it is a quick and awesome read. If you have a more specific question about the period, we can help.

EDIT- after reading mod's comment, it is what is happening locally.

A good place to start is The Cartoon History of the Universe Vol. III. It will give you a quick overview of the religious and political conflicts between Ethiopia and Yemen leading up to the birth of Islam, and the relationships between the various family members that led to the conflicting views on rights of succession.

It's very simplified - it is a comic book after all - but It's a great way to get a broad overview of what was going on if you know absolutely nothing, and it has an extensive and detailed bibliography of primary sources, including recommendations on which ones are an enjoyable read.

Here are some suggestions for further reading...

As for the Quran itself, check out Michael Sells, Approaching the Quran.

For pre-Islamic Arabia, check out Robert Hoyland, Arabia and the Arabs.

For the latest interpretations of early Islamic history and the development of Islam from the life of the Prophet Muhammad through the Umayyads, check out Fred Donner, Muhammad and the Believers.

Edit: Fixed formatting.

There have been a lot of good posts here thus far. I would go further and argue that to really understand the Quran, you need to look at the status and beliefs of the Arabs before Muhammad's revelations. There is a lot of interesting parallels between pre-Islamic Arabian beliefs and Islam, though there are also substantial differences. These differences are easier to understand once you see how Muhammad's revelations were addressing many of the issues associated with Arab society at that time.

I highly recommend Karen Armstrong's biography of Muhammad, it does a great job discussing the issues of tribal warfare, geopolitics, Arab oral culture, pre-Islamic religious rituals, and the process of Muhammad's rise and dominance as a religious figure.

Hi, usually I wouldn't give top level comments, but I think these Yale University lectures will be suitable for your needs. Take a look! The videos are summarized in the description.

http://www.youtube.com/watch?v=cXF7RkgANNo

I have almost finished watching the entire series and I think (by my amateur standards) these are really nice lectures.