The Umayyads did not just complete and consolidate the conquests of the Arabs. They also established lasting forms of governance. They quickly abandoned the practice of having elders come together to appoint leadership, insisting on a hereditary line of caliphs. This alone caused a civil war in the late seventh century, as some of their Muslim subjects rose up, claiming that they had perverted the proper line of leadership in the community. The Umayyads won that war, too.
The major problem for the Umayyads was the sheer size of their empire. Just like other rapid conquests, like that of Alexander the Great 1,000 years earlier, in the course of just a few decades a people found itself in control of enormous swaths of territory. The Arabs had a strong lingual and cultural identity and many of the Arab conquerors saw themselves as a people apart from their new subjects, regardless of religious belief. Thus, while non-Arabs were certainly encouraged to convert to Islam, the power structure of the Caliphate remained resolutely Arabic. As with the Greeks under Alexander, the Romans during their centuries of conquest, and the Germanic tribes that sliced up the western Roman empire, the Arabs found themselves a small minority ruling over various other groups.
To try to effectively govern this vast new empire, the Umayyads took over and adapted the bureaucracies of the people they conquered, including those of both the Byzantines and, especially, the Persians. They created new borders and provinces to better suit their administration and ensure that tax revenue made it back to the capital at Damascus, with the idiosyncratic additional factor of needing to pay an ongoing salary to all Arab soldiers, even after those soldiers had retired.
One change that was to last until the present was lingual. Unlike in the Greek case during the Hellenistic period, Arabic was to replace the vernacular of the land conquered during the Arab conquests. The only exceptions were Persian, which would eventually become the modern language of Farsi (the vernacular of the present-day country of Iran), and Spain, where Arabic and Spanish coexisted until Christian kingdoms reconquered Spain many centuries later. This lingual uniformity was a huge benefit to trade and cultural and intellectual exchange, because one could travel from Spain to India and speak a single language, as well as be protected from bandits by a single administration.
Arabs also followed the patterns of Greek and Roman conquerors by colonizing the places they conquered. At first, they settled in garrison and administrative towns, but they also set up communities within conquered cities. As Arabic became the language of daily life, not just of administration, Arabs and non-Arabs mixed more readily. Arabs also built new cities all across their empire, the most notable being a small town in Egypt that would eventually grow into Cairo. They built these cities on the Hellenistic and Roman model: planned grids of streets at right angles. In the center of each city was the mosque, which served not only as the center of worship, but in various other functions. Mosques were both figuratively and literally central to the cities of the Umayyad caliphate. They were the predominant public spaces for discussion among men. They were the courthouses and the banks. They provided schooling and instruction. They were also often attached to administrative offices and governmental functions.
The Umayyads imposed taxes across their entire empire, even insisting that their fellow Arabs pay a tax on their land, which was met with enormous resistance because, to Arabs unused to paying taxes at all, it implied subordination. By channeling taxes through their new, efficient bureaucracy, the Umayyads were able to support a very large standing army. That allowed them not only to keep up the pressure on surrounding lands, but to quash rebellions.
The Umayyads oversaw a tremendous expansion in trade and commerce across the Middle East and North Africa as well. Muhammad had been a merchant, after all, and the longstanding commercial practices and regulations of Arabic society were codified in Sharia law - in that sense, commercial law was directly linked to religious righteousness. Likewise, even from this early period, the caliphate supported maritime trade networks. Muslim traders regularly sailed all across the Mediterranean, the Persian Gulf, the Indian Ocean, and eventually as far as China and the Philippines. In waters controlled by the caliphate, piracy was contained, so trade prospered even more.
One effect of Arab seafaring is that Islam spread along sea routes well beyond the political control of any of the Arab empires and kingdoms to come; today the single largest predominantly Muslim country is Indonesia, thanks to Muslim merchants that brought their faith along the trade routes. By the time European explorers began to establish permanent ties to Asian kingdoms and empires in the sixteenth century, Islam was established in various regions from India to the Pacific, thousands of miles from its Middle Eastern heartland.