Afro-Eurasia is a vast landmass, including everything from the southernmost tip of Africa to the far northern areas of Scandinavia and Russia and from China in the east to France and Britain in the west. This area was home to many different cultures in the late sixth and early seventh century as Islam arose. This answer will cover several of the major cultures of the period and region.
Perhaps the most notable event in western Europe in this period was the collapse of the Roman Empire. The area from Italy north to Hadrian's wall in Britain and much of North Africa had been under the sway of the western, Latin-speaking part of the Roman Empire as a result of conquests originating in the Punic Wars. With the fall of the city of Roman, and barbarian conquests of former Roman territories, this area fell into what are sometimes called the "dark ages", a period where much of Europe was split into various smaller kingdoms, marked by a decline in culture and prosperity.
In the Greek east, the Roman Empire survived in the form of the Byzantine Empire, Greek speaking and Christian but still following many Roman traditions and laws. Although the Eastern Schism occurs substantially later, in this period we can see the beginnings of the divisions of Christianity into Orthodox and Roman, as the Greek and Latin traditions began to diverge.
In Persia, the Sassanid Empire continued earlier cultural traditions, following the Zoroastrian religion. It was the main rival of the Byzantine Empire. Its eventual defeat by the Arabs led to the rise of Islam in Iran, replacing Zoroastrianism.
China was flourishing under the Tang dynasty in this period; it was a powerful and stable dynasty renowned for its art. The Gupta Empire was flourishing in India, where Buddhism and Hinduism were the major religions.