Slavery continued well into the early medieval period in Europe, but with one major caveat: the Christian church forbade the owning of Christian slaves by other Christians. The Roman Empire's reliance on slavery was built on two contrasting pillars: the Pax Romana and the Roman war machine. Rome's wars of conquest, its pacification of provinces and its frontier wars provided a steady stream of captives to feed the demands of the Empire, and the internal stability and vast logistical networks of the Empire allowed those slaves to be distributed throughout the Empire in vast numbers. As the Empire declined in the fifth century, so too did its logistical and trade networks. As provinces fell back on local forms of government and populations were no longer able to rely on imported food - or imported labour - populations abandoned the slave-driven Latifundia complexes and reverted to variations on subsistence agriculture and more local power relationships.

Slavery continued in a number of contexts and in a few major avenues. The Holy Roman Empire predated upon the Pagan tribes to the east of their frontier, the English made slaves of the Welsh and Irish, and the Vikings made slaves of, well, a lot of people before they too Christianised and focused on the Irish and the Slavs. Slavery was also carried on around the Mediterranean between Christian and Muslims, and across the frontiers of Iberia. The central contexts necessary were some kind of frontier or conflict, and a difference of religion. By the time of Domesday Book in 1086, 1 in 10 households in England was listed as consisting of slaves, and that was after the practice had been in decline for some time due largely to moral pressure from the Church. By the mid-twelfth century, slavery does seem to have largely disappeared in England, in part because it was becoming increasingly difficult to find viable non-Christian populations on which to prey, but also largely due to religious pressure to end what William of Malmesbury called a 'vile' practice.

Pelteret's Slavery in Early Medieval England is a useful read on the subject.

I answered partly on the topic recently there : as there's no clear consensus how exactly serfdom appeared as a set of statutes, it's not to be taken as a definitive answer, there's more to be said.

As it seems to have happened in the VIIIth to Xth centuries, it doesn't seem to have been a transformation of slavery into a semi-servile social class, but rather the result of new social polarization allowing or forcing social shifting on people. It's not a given that people had a clear distinction between slave and serf at this point, at least not the one we have.

Slavery itself did continued to exist at least until the Late Middle-Ages for western Europe (more or less in the same time serfdom in the strictest sense disappeared) and it coexisted along various intermediary statutes between servile and free peasantry appearing in the Late Empire while the existence of Mediterranean slave trade, supplied by western kingdoms, fueled the maintain of slavery there too.

That being said, being a slave in the VIIth century wasn't necessarily the same than before : intermediary statues blurred the distinction between free/non-free (until it more or less settled in the general disfavor of semi-free peasantry), and "areas of freedom" probably punctuated slavery as a statues (e.g. slaves living "as coloni")