A student of mine once wrote on an exam that "surfs were tide to the land." I like the maritime motif, but um, no.
A serf, servus in medieval Latin, was someone who was not legally free yet not a slave. We now have trouble with such distinctions, but both ancient Rome and medieval Europe were fine with it. Slavery means that someone is assumed to have no will of their own and are thus required to obey all commands, no matter how stupid, arbitrary, or harsh. A slave can be bought and sold like an animal or any possession.
Serfs could not be bought and sold, and although they might have very heavy obligations, they were not subject to arbitrary commands. They were still, however, assumed not to have complete freedom of will, so they could not give testimony in court or join the church, both of which required free oaths. A serf was born into serfdom, under the authority of a "lord of the body." Because they could not be sold, they could not be sent off their land (this is what "tied to the land" meant--it is sometimes described as a horrible condition, but in fact it was a protection).
Serfdom began in the sixth century, at the end of the Roman Empire. Although the period is very badly documented, so the details are not clear, what appears to have happened was that as the Roman army stopped conquering and thus stopped bringing home hordes of new slaves, landlords became worried that their agricultural slaves would die and not be replaced, and they would have no one to do the work. Hence, they allowed their serfs to marry, establish families (thus creating new workers), and have their own little plots of land so that they could support themselves, rather than having to be fed by the landlord.
Helping hurry the end of slavery was the spread of Christianity, which had always preached that all humans were equal in the eyes of God. It was considered a "good deed" to free one's slaves. Household slavery lingered in Europe until the eighth or ninth century, but agricultural slavery was gone by the seventh century, replaced by serfdom. And of course there were always free peasants as well as serfs, on whom see more here.
Serfs were assumed to be born into serfdom; it was inherent in their bodies. They owed heavy rents, especially labor dues, that is a certain number of days a week they were required to work on their lord's land rather than their own, but these rents were fixed, and the lord could not raise them. Serfs also had other restrictions, such as who they could marry and who could inherit from them, and in some places they had to come before their lord once a year with a rope around their neck and a penny on their head, to indicate their subjection.
I will discuss the end of serfdom in a later post. In the meantime, just don't call it feudalism.