Our perceptions of what Ancient Roman society and daily life were like tend to focus on the towns and cities, since that's where we have the best evidence from (most Classical Roman authors were from the urban elites, we have unparalleled archaeological evidence from Pompeii and Herculaneum etc). Yet like in all pre-industrial states, the majority of the Roman Empire's inhabitants would have been rural dwellers and, to my rather limited knowledge of ancient history (I'm more of a medievalist), the slave latifundia system of agriculture only predominated in Italy, Sicily, Greece and the West Coast of Asia Minor. Thus most of the Roman Empire's inhabitants from lowland Britain to Egypt must have been peasants of some sort. What was their material culture and standards of living like? What were their family structures like? What relationship did they have with senatorial aristocratic landowners? What sort of settlements did they live in?
In attempting to answer this question I will be focusing primarily on the era of the late Empire since my knowledge of the early empire and republic periods for the peasantry is limited, and primarily on the difference between the peasantry on the social level between East and West.
The first step is to differentiate between the two basic types of peasantry (though this is not meant to imply that there were only two types at all times). On the one hand we have the servi; people who were not free and had no rights under the law, and the coloni; people who were basically understood to be tenant renters of land owned by landlords.
The servi are not to be understood in this context as slaves. By the time of the late Empire (in the West) the plantation slavery of the classical Roman Empire in Italy and Greece had fallen into general disuse. Historian Chris Wickham notes that using the term ‘slave’ as a synonym for servi is misleading. Servi were tied to the land in a way similar to that of medieval serfs, but they were not exactly slaves as we might think of from the earlier empire.
The East seems to have seen more landowning on the part of the peasantry than the West, due to the fact that there were fewer of the large landowning estates. Places such as Italy, Africa, and parts of Gaul in the West were much more known for having large swaths of land owned by wealthy people and thus, not peasants. The East/West split was not uniform, though. There were rich landowners in the east and there were peasant landowners in the west. But the surviving records and material culture seem to point to a general trend such as has been described.
You can find details about the Romano-Egyptian village of Aphrodito and some if its inhabitants, which supplies us with a tremendous amount of knowledge about later imperial villages and peasant life, here:
https://www.lib.umich.edu/snapshots/Dioskoros/dioskoros.html
Surviving tax laws also make a distinction between coloni who actually owned land and paid taxes directly to the tax collectors, and coloni who owned no land and paid taxes through their landlords. These were known as adscripticii. These latter peasants were really more similar to the servi in that they were dependent on their landlords to a greater degree. Servi did not pay tax at all; it was paid for by the lords.
In Corpus jurus civilis, a 6th century collection of Roman laws, the emperor Justinian notes that there really was little difference between the servi and the adscripticii. In an earlier time these would likely have been considered slaves, though the changing times had already created a distinction between them and the slaves of earlier times.
In many cases what differences there were varied by region. Egypt generally saw shorter leases, more peasants actually involved in land ownership, more wage labor, and a general rarity of unfree rural workers. In Italy, meanwhile, the ancient stronghold of landed estates worked by slaves, a far larger proportion of tenant workers were unfree.
In the West, it was far more common for peasants to live in towns or cities. Eastern peasants lived more commonly in rural villages. This is particularly true of Egypt though it can be said for much of the Greek speaking empire in general. Owners and their tenants usually lived in these villages together and thus, peasant society was typically more autonomous than in the West since the administrators of the land lived on their land along with their tenants, and not in a town or city many miles away from the land they owned.
The Western Empire, by contrast, shows a landscape scattered with rural villas and great estates of the wealthy landed elite and isolated farms. Land was usually identified by its owner and estates typically had their own names. Since these estates were so far-flung, they were not the village communities of the East, where peasants lived side by side with their landlords in organized settlements. In general, the gap between rich and poor was far wider and more pronounced in the West.
All of this is to say that depending on where one lived during the late Empire, the chance of....not exactly upward mobility, but a certain autonomy and social freedom certainly existed for the peasantry. Eastern peasants thus worked the land they owned or rented, lived off the food they harvested, gave surplus to their landlords, and paid tax to the state. In some ways it is easy to see a dim resemblance of the modern “middle class”, though obviously the two circumstances are very different.
The scarcity of such conditions in the West, it might be argued, helped contribute to the myriad problems that eventually caused the empire to end nearly a thousand years sooner than that in the East.
Further reading: Inheritance of Rome- Illuminating the Dark Ages (400-1000), Chris Wickham, 2009