Welcome to Reddit! Briefly, writers in the Western Roman Empire used Latin. (The Roman Empire in the East, which did not "fall," is another question altogether.) Their word for slave was servus, which kept getting used throughout the middle ages. In the mid-1800s, historians (such as Karl Marx) noted that a servus under the Roman Empire lived very differently from a servus living in high medieval England. To help make sense of this difference, they started calling the classical Roman servus a slave and the late medieval servus a serf.

What's the difference? A slave, in this narrow sense, is someone who belongs to another person, whereas a serf is someone who belongs to the land. The land might change ownership, but the serfs stay in place. Historians have spent a long time debating over when the transition from slave to serf happened, at it's been pretty inconclusive. It's also complicated by a bunch of other words with similar problems: sclavus (an enslaved Slav) and ancilla (a slave-girl or maid) are two.

Leaving the debates of historical journals and returning to the world of late antiquity, it looks like slavery remained significant for centuries. How significant is impossible to tell, since we don't really have statistical evidence for much of anything. Michael McCormick (Origins of the European Economy) thinks the slave trade sparked the European economy in the 800s. Chris Wickham (Framing the Early Middle Ages) doesn't find much evidence for slave production at all. So it's hard to tell.

Two great starting points, and I can recommend more if you've got further questions or are looking for something more specific:

• McCormick, Michael, “New Light on the ‘Dark Ages’: How the Slave Trade Fuelled the Carolingian Economy,” Past and Present 177 (2002), 17-54.

• Stuard, Susan Mosher, “Ancillary Evidence for the Decline of Medieval Slavery,” Past and Present 149 (1995), 3-28.

That depends on a few things my friend, what do you consider to be the date of the fall of the Roman Empire? Are you wondering about the period after September 4 476 A.D.? How many years after this 'official date' of the fall of the Roman Empire according to Gibbon? Are you interested in 476-1050? Are you interested in Italy specifically or all the regions of the Western Empire, or Eastern too?

Slavery took a very wide variety of forms depending on the time period and region you're referring to. In terms of Italian states that followed the Empire there is some evidence of the abolition of slavery by the Doge of Venice (960).

But that definetly did not last as Venice and Genoa in following centuries conducted a huge slave trade buying Ukrainian, Russian, and other Slavic people and selling them in the Ottoman Empire, Egypt, and North Africa.

With the advent of Islam a whole new line of thinking emerged. Muslims 'prohibited' enslaving Muslims and Christians 'prohibited' enslaving Christians. Neither side adhered to this principle. According to Salvatore Bono's "SCHIAVI IN ITALIA: MAGHREBINI, NERI, SLAVI, EBREI E ALTRi" slavery was abolished in the Kingdom of Naples more than three times: once in the 1200s, and twice in the 1500s.

Generally slavery persisted, but as populations declined and populations became interconnected it became harder to pursue with neighboring populations. The Romans under Caesar were able to enslave 53,000 Gauls because of asymmetric technology and power. Post Roman Western Europe was a different setting, the populations and political leaders were linked economically, dynastically, linguistically, and religiously. If one Frankish warlord mass-enslaved his neighbors there would be steeper consequences.

For this reason slaves tended to be acquired from places very far away. Egyptians, for example, didn't enslave Tunisians or Syrians for the same reason, they were linked to these populations and while war was acceptable, enslaving coreligionists was very taboo. However, acquiring slaves from, say, infidel populations in France or Southern Italy was more acceptable.