First of all, it's a bit difficult to speak of the fall of the Roman empire as the imperium had been split up after the death of emperor Theodosius in 395 AD. After that, the Western Roman empire fell, but the Eastern Roman empire remained more or less intact for several hundreds of years to come and ceased to exist only after Sultan Mehmed II. Fatih's conquest of Constantinople in 1453. But let's keep an eye on the Western part.

At the end of the 4th century AD, the Huns started migrating westwards, coming from central parts of Asia. In between them, we find various Germanic tribes or peoples or groups of humans (modern boxes like »tribe« or »people« are actually quite misleading) which then started migrating: Firstly due to the Huns pushing from the East, but secondly also attracted by the relatively richer and better organised, thus superior Western Roman Empire to the West of them. This Migration Period is fairly chaotic – some groups came with »all men« (hence the French or Spanish name for Germany: »Allemagne«/»Alemania«), others are hard to trace. Some stuck to their Germanic traditions, others can be considered Christians, though they'd usually follow Arianism rather than the Roman Catholic traditions since Ulfila's mission.

The three groups you'll want to trace (unless you're interested in trivia e.g. why we speak of »vandalism« or how the Istanbul football team Galatasaray got their name), are the Goths – subdivided into the Greuthungi or Ostrogoths and the Thervingi or Visigoths – and the Franks.

The Osthrogoths had a brief and peaceful kingdom in Italy in the 5th century AD; first tolerated by Constantinople as they'd asked the Osthrogotic king Theoderic the Great to fight Odoacer after he'd crowned himself as rex Italiae. The Osthrogothic kingdom didn't last long and would eventually vanish into chaos.

The Visigoths have been around for slightly longer; first as foederati, meaning auxiliary military troops during times of Western Roman weakness. The Roman aristocracy would choose to rely on »barbarians« for their defense; since the two parts of this cohabitation were kept apart by a multitude of differences – different legal systems, different religious beliefs, different social groups – animosities grew over time despite a process of gradual, slow acculturation. Eventually, the Visigothic kingdom known as »regnum Tolosanum« (empire around Toulouse, modern day France) migrated towards the Iberian peninsula forming a second empire there, the so called »regnum Toledanum«, named after the city of Toledo in today's Spain.

The Franks originated from what is nowadays known as Brabant – essentially the South and Southwest of the Netherlands and the Northern »hump« of Belgium. Their first influential dynasty were the Merovingians; keep an eye especially on Childeric who forged ties with the Western Roman empire during the 5th century AD and then on Clovis I who did what no German king had done before: He converted to Catholicism. This move resulted in acceptance by the Gallo-Roman nobles and hence a stable reign which, step by step, became bigger and bigger until it was the dominating power in what we know today as Northern France, the Benelux countries, and Western Germany. It's around this time, roughly 500 AD, that sources begin to to show up in greater numbers again. For instance, Clovis' time is decently covered by the works of Bishop Gregory of Tours (needless to say that the Catholic bishop was fervently in favour of the first king to adopt catholocism, still the sources are okay in quality). Later in time, the Merovingian dynasty was replaced by the Carolingian dynasty after having received the blessing from the Pope in Rome who, on Christmas eve in 800 AD, crowned Charles the Great as emperor.

This is how Germany – as the Eastern part of the Frankish empire – inherited the title of »Holy Roman Empire of German Nation« (which ceased to exist only in 1801... well, partly due to the belief of the »translatio imperii«), but also how France – as the Western part of the Frankish empire – got its name. Aside of that, especially the Carolingian renaissance has left countless traces which we continue to use until today. To name only one thing: small letters!

After that, Slavic tribes began migrating into the Eastern and Southeastern parts of Europe, Turkic tribes came in two shifts – one merging with the Southeasternmost slavs into the people we know as Bulgarians, the second one as actual Turks forming the Ottoman empire; but I've spoken about Mehmed Fatih earlier already. Another minor migration was that of the Hungarians to Central Europe and the Finns (more precisely: Finns, Tavasts, and Karelians) and Estonians to Northeastern Europe; both of these groups were settled at around 1000 AD when the Middle Ages in Europe were already in full swing.