Peasants are people who live on the land and depend for their survival primarily on the cultivation of this land. In pre-industrial societies they make up the majority of the population, but there is little place for them in a society that has adopted the mass production of food. The transfer to modernity depends upon the liquidation of the peasant economic and social structure, and there have been many movements in the twentieth century which resented this destruction and sought a return to a usually romanticised past.
The reason the market has such a corrosive effect on peasant society and economics is that your average peasant does not aim at profit maximisation, but at risk minimisation. As the peasant household is a unit of consumption as well as of production, peasants face a constant threat of dying if they fail to produce enough to feed themselves and their family. Peasant societies are then, quite naturally, geared to minimise the risk of any member dying of starvation. This is the importance of land ownership in peasant society - land is the best insurance that you and your family will not starve.
Hence peasants have a peculiar relationship to the market, which has led some to comment that they do not behave as neo-classical economics predicts they should. This is misleading. The only resource peasants have in abundance is their labour (and land if they are lucky), and neo-classical theory would seem to predict that they would dispose of this labour in a way that brought the largest marginal utility (that is, the greatest return for the expense of 'units' of labour). However, it seems that what peasants value most of all is stability and security, not high profits: they are risk averse.
This can be seen in the social structure of peasant society. The traditional hierarchy is smallholder at the top, followed by tenant, followed by landless labourer. But these distinctions are not based on gradations of wealth; they are based on gradations of economic security. A smallholder might not be as rich as a tenant with a large plot of land, but his economic security is in the hands of God and not another man, as only a bad harvest and not the decision of a landlord to raise rents can ruin him. Similarly, wage labourers might do very well out of the cash economy, but they are exposed to the vagaries of this economy and of the labour market. Hence, to be a landless labourer is to be in the most insecure position in peasant society.
Peasant institutions, cultural mores and social interactions are based on the principle of a right to subsistence. The rich are generally expected to be benevolant and the poor to be taken care of - a class of starving peasants would be an existential threat to the rich. This is what leads to the generally very good networks of social support in peasant societies, and leads to their romanticisation by the opponents of modernity. As the state began to make more demands on peasants and their resources, their reaction was based on a consideration of this moral universe. The state's claims would be evaluated not based on how much it demanded, but how much it left. A demand for twenty units of corn in a bumper harvest year would be seen as less oppressive than a demand for ten units in a disastrous year.
The increasing demands of the state, especially forceful ones in Communist societies, and the market tend eventually to destroy this peasant society. As land tends to accumulate to large landowners due to favourable economies of scale, peasants move into the cities and the cash economy. This destruction of the traditional way of life and method of subsistence for a large part of the population and the transfer to economic uncertainty is the cause of the greatest tension on the path to modernity. This is especially so where no welfare state is provided to continue to uphold the right to subsistence.