This a pretty huge question. In very general terms I can touch on a few points and perhaps expand on them if a specific area is of interest.

The Arabs existed on the periphery the Roman and Persian worlds. The word itself refers to their propensity for a nomadic and desert existence. Significant kingdoms of the Arabs existed as client states of the Romans and Persians while Arab traders and settlers permeated both empires, particularly in the adjacent and fertile lands of Mesopotamia and Syria.

The settled cities of western Arabia seem to have been important for trade, although the trade of what and how important this trade was are a matter of dispute. While nomads could traverse the interior the principle cities were along the western coast extending from around modern Aden in the Yemeni coast and extending along the trade routes to modern Jordan and including Mecca and Medina.

Religiously, and befitting a peripheral culture, Arabia was a great mix. The traditional religion was polytheistic but Arabia was also host to the Jewish Kingdom of Himyar in Yemen. The Lakhmids, a Persian client kingdom in the north east were predominantly Nestorian (Eastern) Christian while the Ghassanids, a Roman client kingdom, were often monophysite and so were heretical in the eyes of Rome but less so than the Nestorians. Mecca had an idolistic religious culture focused around the Kaaba and included holy religious months of truce, both of which were re-oriented in Islam.

Militarily, the client kingdoms I mentioned above grew in importance during the great Roman-Persian wars only to be abandoned by both empires leaving, as a result, huge client armies that were deeply familiar with the Roman-Persian way of war but no real allegiance to either state. Tactically, camel-borne nomadic combat was the norm.

Culturally Arab society was almost entirely illiterate but deeply valued oral forms of "literature," particularly poetry, the epics of which have been preserved. It's difficult to say how accurate this may be but early Islamic writings including the Quran denounce pre-Islamic Arabia as a profoundly decadent society that was full of drunks and murdered children and all that. It's difficult to judge historically because A: we have virtually no other sources and B: things infanticide seem to have been regarded as somewhat normal in the antique world, so its hard to say whether or not Arabia was in any way abnormal in its morals.

I guess to conclude I would just emphasize that although the Arabs were often contrasted with Rome and Persia in a "battle of the civilizations" sort of way (Nomads vs. Settled, Christianity vs. Islam, Antique vs. Medieval) Arab society was very much a part of the late Antique world which is why the Arab/Islamic conquests aren't quite so much as a catastrophe to civilization as the loss of territory may at first seem. I would argue that this is arguably even more the case for the Arabs than for what I understand of the reappraisal in recent decades that's been afforded to the "barbarians" of the western-empire.

I would also highlight that even during the Arab/Islamic conquests, even though Islam had united the Arabs, it had not actually converted them, so that large portions of the Arab armies in the "Islamic" conquests were actually still Christian. Islam did not become a universal religion of the Arabs, let alone for non-Arabs, for at least another century after Muhammad.

Sources: I would recommend two books by Robert Hoyland Arabia and the Arabs and In God's Path as being relevant to your question. In reference to the debate on Mecca's place as part of Arabian trade routes and by extension some of the problems relating to Arabian illiteracy and the lack of contemporary sources, see Patricia Crone's Meccan Trade and the Rise of Islam.

Although it by no means focuses solely on the Arabs, for a broader understanding of Rome, Persia, the Arabs and the greater cultural context of the age I highly recommend Peter Brown's The World of Late Antiquity particularly for its reappraisal of the Arab position in the antique world.