Medieval and early modern Western Europe is a huge tract of history. I can only really address the concept of "peasant" as it's used to describe conditions in the early Holy Roman Empire.

During this time period, peasant was definitely both a social and an economic distinction. You were a peasant if you were primarily an agricultural laborer and people didn't recognize you as a noble. You could be a very rich peasant, in terms of land your family owned outright, you might even have some other people paying you rents, but if you didn't act the part of a noble (you didn't have any horses, you didn't give gifts, you messed up etiquette), you weren't given noble status by peers. And it was their opinions that mattered.

There were a variety of ways to be a peasant. You had free tenants, you had various sorts of leases, and you had people who had sold you their land (or gifted it to a church) when they fell on hard times but maintained the right to live there and work it because you gave it back to them, sort of, after they sold/gifted it to you. Peasants also owed military service. But a peasant usually didn't have a horse.

What happened before this time is important. There was a reorganization of peoples the Romans had classified as Germanic following the decline of the Roman Empire, when a bunch of peoples were moving around settling into the gaps left by the collapsing western Roman Empire. When you get to about the 8th and 9th centuries, a lot of people have settled into the old power positions, and they want to stay there. But when you want to stay somewhere, then you have problems with how to defend your lands against other people who want it.

A horse soldier is vastly superior to a foot soldier in terms of mobility and the force he can put behind his weapon. But horses are expensive, especially the sort of heavy horse you'd use as a war horse. To this end, you wanted to spread these costs around and give people a reason to want to help you keep unwanted settlers off your lands, instead of just run off when they showed up.

So if you had land to spare, you would give someone the right to use the land with conditions attached. Usually those conditions included making productive use of it so that your family could afford a weapon, keep a horse you maybe gave them as a gift, and send a man to help defend the land when necessary. If you neglected those duties, you might have your land taken back.

If you weren't an aristocrat (keeping in mind that both religious and lay authorities owned and managed lands), and you were involved in agriculture, you were some variety of peasant.

Nobility didn't explicitly or solely rest on the possession of land. You also had to have the excess wealth to be able to give gifts and to obligate people socially. Land transactions are well-recorded because land was so important, but you can also see the importance of other gifts in sources like the writings of Theodulf of Orleans, or Charlemagne's letter to Pope Hadrian that he sent along with some gifts. A noble was a person who had noble character and was recognized by his peers. This entailed giving gifts of horses, weapons, and clothing to the people in your entourage, giving gifts to peers or superiors, and giving gifts to the church.

Sources: Living in the Tenth Century: Mentalities and Social Orders; Morality and Masculinity in the Carolingian Empire; Medieval Europe from 395 to 1270 (extremely dated, but with fantastic source footnotes); The Languages of Gift in the Early Middle Ages.