Modern medievalists (post-1980s) by and large no longer use Feudalism as a concept to describe medieval society. Here's a good discussion on that point.

The structure of peasant relationships to their lord (or abbot) was a complex, fluid one that changed a lot over the Middle Ages as peasants and lords jockeyed for economic power. It was also different in different regions. This would look different in England than it would in the south of France, which was different still to the structure in Italy. There were also cities, which didn't function like rural areas at all. We know the most about England because we have by far the most and the earliest records from there. This is a rough explanation, based mostly on England.

The basic way that peasants were defined was whether they were free or unfree. This is a little confusing in a modern context (especially American) when freedom is explicitly associated with slavery. Unfree peasants were not slaves. In fact, in some times and places there were both unfree peasants and slaves living simultaneously, side by side. What free or unfree meant was how much the person owed the lord. This could be in land rent, obligations, special taxes, right to get married without permission, etc. The peasant farmer had a strong incentive to become as free (of obligation) as possible, so that more and more of his labor would benefit himself. The Lord had an incentive to argue that more and more of his peasants were more and more unfree, so that he was owed more in obligations.

These obligations were both in taxes/rents, such that a tenant family might owe a certain amount of money (often paid in kind) and gifts-in-kind (small livestock like capons, produce) based on their free or unfree status. The freeholder would owe far fewer fees and gifts, while the totally unfree peasant owed constantly.

On top of this, the tenants would owe work to their lord. This was calculated out to how many days of work were owed, and those days had to be spent on the land that specifically belonged to the lord, rather than the land that the peasant rented from the lord. If the Lord had more work than he had obligations from peasants, he would supplement the owed labor with paid labor. This could come from multiple sources. Servants/farmhands who made a living working for the lord, paying the peasants that already lived on the lord's lands for extra work, and migrant laborers who picked up work. It's really hard to know much about who these migrant laborers actually were or where they came from, as they didn't really leave behind any sources.

As a follow-up to Ziegler, you may be interested in John Aberth's From the Brink of the Apocalypse: Confronting Famine, War, Plague and Death in the Later Middle Ages. It's about the cultural implications of the massive upheavals of the early 14th century.

Sources:

RH Hilton, The English Peasantry in the Later Middle Ages: The Ford Lectures for 1973 and Related Studies

Barbara Hanawalt, The Ties that Bound: Peasant Families in Medieval England

Kathleen Biddick, *The Other Economy: Pastoral Husbandry on a Medieval Estate

Richard Britnell and John Hatcher, eds., Progress and Problems in Medieval England: Essays in Honor of Edward Miller

Paul Freedman, Images of the Medieval Peasant

I also have a lot of case-studies of different manors and villages, if you'd like references to those. They're nitty-gritty social histories, though!