I've been waiting for ages for a question like this to come up.

The truth is that it varies wildly based on what part of Europe we're looking at. The changes in the post-Roman world were most pronounced in the Western half of the former Empire. Straight off the bat, the most important distinction between the Roman world and the Early Medieval one was the change in the levels of urbanization, especially in the West. Before the fall of the Roman West, cheap grain from North Africa collected by taxation allowed the Romans to inflate the size of their cities. After the Vandals took North Africa and cut off the Roman grain routes, the population of the major western cities plummeted dramatically. According to Chris Wickham's The Inheritance of Rome (a great book on this whole time period), the city of Rome suffered a population drop of about 90% from the late fifth to the end of the sixth centuries.

Some regions in Western Europe managed to remain fairly stable, particularly in the case of Ostrogothic Italy and to a somewhat lesser degree in Visigothic Spain. In these regions the Germanic conquerors took control of territories that still had large populations of educated, upper-class Romans such as Theodoric the Great's secretary, Cassiodorus. With large numbers of educated men, the Gothic successor states to the Western Roman Empire were able to do a fairly good job of maintaining the bureaucratic structures that had they inherited from the Romans. With Roman government still in place, things like laws and education and taxation could still be meted out in an effective manner, and because of these life did not change considerably for the average person living in Spain and Italy. Perhaps the only really considerable difference was in religious matters. The vast majority of the Germanic peoples were Arian Christians, a heretical sect of Christianity that did not conform to the idea of the Holy Trinity as established by the Council of Nicaea. Most of the people who were already living the parts of the Roman Empire conquered by Germanic Barbarians were Nicene Christians, so the conflict with their rulers often came down to religious matters. However, this was a primarily urban phenomenon and it is unlikely that Roman peasants ever had any real contact with their Gothic overlords. After Theodoric's death in 526 matters in Italy began to go down hill, and when Justinian and the Eastern Roman Empire began their grueling reconquest of the peninsula Italy seems to have been "worn out," so to speak.

Merovingian Gaul and Vandal North Africa provide somewhat different cases. The Vandals largely did away with the remaining civil service and were far more zealous about promoting Arian Christianity, but the Vandals were rather quickly wiped out in the early stages of Justinian's reconquest. The Merovingian Franks in Gaul salvaged what they could of the Roman infrastructure they had inherited, but the constant infighting after the death of Clovis left Gaul in a sorry state. By the beginning of the Carolingian period, there were few major cities and greatly reduced trade. The living standards of the average Gallo-Roman living in Frankish Gaul were probably significantly lower than they had been during the years of Roman rule. Of all the former provinces of the Roman west, Britain was arguably hit the hardest. Britain had always been on the edge of the Empire and probably did not have as strong a class of educated Roman aristocrats as their counterparts across the channel. At some point before the beginning of the seventh century, Germanic groups such as the Jutes, Angles, and Saxons crossed the English Channel. Written records in Britain from 450 to 650 are virtually non-existent and are so scant that we're not even sure of when the Germanic peoples showed up in the first place, although they probably came gradually starting as early as the late fifth century.

In the east, the Roman Empire managed to remain in fairly stable condition. Cities such as Alexandria, Constantinople, and Antioch remained fairly large, although populations shrank over the given period due to outbreaks of plague. Constant war with both the Germanic successor states and the Persians combined with recurring bouts of plague weakened the Roman East to the point that the Arabs were able to conquer large swathes of Egypt and the Levant with relative ease.

Any questions?

Sources:

• The Inheritance of Rome by Chris Wickham, a great overview history of the Late Roman and Early Medieval World and a must-have for any historical library.

• The Goths by Peter Heather, a first-rate history of the Gothic peoples that is as straight-forward to read as it is factually interesting.

• The Franks by Edward James, an excellent and succinct (perhaps a little too succinct) history of the Franks, particularly during the Merovingian period.

Edit: Wanted to add some descriptions and I also forgot a source:

• The Anglo-Saxons edited by James Campbell, an outstanding illustrated history.

So there are a lot of books on this subject. I would recommend The Early Middle Ages 400-1000, a Short Oxford History of Europe, edited by Rosamond Mckitterick. Its a good intro to all aspects of the era, from political to economic to social, and all of the historians in that book are at the top of their field.

As I'm sure you can tell, it's quite hard to sum up and generalize 300 years of very pivotal European history into one short post. And as I'm sure you can guess, the idea that Europe was sent back to the dung ages of an apocalypse like wasteland is a bit overdramatic.

Rather than get into the nitty gritty of whether or not people became superstitious idiots or everything continued on like it did in the past (except more Christian) let me just say that a very important concept to understand for the time after the fall of the western Roman Empire, is this idea of intense regionalization.

The Roman Empire at its height was a multiple continent spanning state that had a high degree of uniformity in its culture and economy across extremely diverse areas, a uniformity in trade and culture that wouldn't be recreated until the early modern era (some say even later with industrialization/westernization/"modernization"). The collapse of the empire meant that uniformity broke down, and each former region of the ex-empire went their own way in adaptation to the harsh new "de-globalized" world. Some survived better than others in some ways, others were worse in other ways.

The bottom line is, it was not all the same, and the assumption that "the dark ages" were uniform in some way, is a modern attempt to legitimize itself as a period of technological progress in contrast to the past as an "other" of bleak ignorance.

Times were shit, for a lot of people, but not everyone. Frankly this could be said for any period, it just depends entirely on who you associate with in the past, and whether you think they were deserving of shit treatment. For example, if you associate yourself with an educated urban bourgeois, you're going to lament the collapse of urban culture after the fall of Rome. But if you associate yourself with say a people rising up against an oppressive autocratic state, you might associate with the barbarians, in some kind of narrative about the struggle for freedom and national identity.

We're actually going to have a "Dark Ages" AMA on Saturday March 8. If you have more specific questions, feel free to pop in and ask a few.

Empires of Faith, by Peter Sarris, is probably the best book I've had the occasion to read on the period (Chris Wickham's Framing the Middle Ages is also great, but its heavy focus on economy and society may be a little overwhelming). It has the merit of articulating the fate of all the parts of the Roman Empire, from Britain to the Persian border, while still retaining narrative parts that allow to make sense of his interpretation (something Wickham usually does not do, which makes him a better second or third read on the question).

However (even though it is of course impossible to answer your question completely), I would like to emphasise that the “fall of Rome” was not the end of Roman structures everywhere. As /u/bitparity underlined, regional variety is everything in this period. Roman Britain, in modern terms, fared the worse; Merovingian Gaul retained Roman features (like taxes) for the 6th century at least, and the Eastern court was apparently pretty “civilised”; Vandal Africa is known mainly by gorgeous mosaics and poetry. It is also important to remember that Eastern Roman emperors reconquered, over the course of the 6th century, Italy, Africa, and a part of Spain. Parts of Italy belonged to the Empire for a very long time: Constans II (emperor from 641 to 668) actually died in Sicily, and the Catepanate of Italy was not lost until the 11th century. Of course, after the 7th century, the Eastern Empire became very much unlike the ancient Roman Empire; nonetheless, it clearly shows that the period was in fact far from anarchy (in fact, if you want to find anarchy, the 5th century is probably a better choice. The 6th century, on the other hand, is the beginning of a new order).