I only have a vague idea about this, but the timeline I have in my head regarding slavery in Europe was that it was a bustling institution during the Roman Empire, faded away going into the Middle Ages, and then came back again in a different, racially-based form going into the early modern era. Did it always exist in some form during all that time, but at lower levels? Why did it go away? Am I just thinking about it all wrong to begin with?
Interestingly, some scholars (I think Lo Cascio is the main one now) have argued that Roman slavery was basically on the decline from the mid first century CE, being out competed by strategies of tenancy and rent seeking.
I talk about this extensively at this link.
However, in a nutshell, it depends entirely upon your definition of slavery.
/u/Tiako's argument is on the basis of slaves operating under "plantation slavery", in which an unfree worker has his labor directly controlled and directed by a lord. However, this means that the lord also needs to pay for people to constantly watch over the slaves lest they escape, as well as feed, clothe, and house them. This kind of slave scheme only works efficiently when there is a steady supply of slaves as well as a stable economy, both of which vanished in the Roman Empire by no later than the 3rd century.
The second kind of slavery is simply legal unfreedom. This lasted far far past the Roman Empire, and was prevalent well into the middle ages, although blurred by the fact that it was hard to distinguish an unfree tenant farmer from a free tenant farmer on a day-to-day basis. Tenancy is basically where the lord takes a percentage of the crop surplus, but the worker still controls his own labor as to what to plant. Because he has that control, he's less likely to flee, and the lord will always take something, albeit at a percentage level.
And of course, the third kind of slave is simply a war captive. And this, well, I think has lasted straight into the early modern era. Certainly the Ottomans were busy with slaves of this type in their galleys.
Also you should keep in mind that though we have distinctive words in English for slave, serf, and servant, in the latin records of late antiquity and the medieval world, this wasn't always the case (e.g. servus, although I'm well aware there are a wide assortment of denotations for specific types of slaves).
Which gets to my original point, what happened to slavery in Europe depends entirely upon what you define as slavery.