I'm going to assume that you mean the fall of the eastern part of the empire in around the year 500, and not the fall of the eastern part in 1204 or 1453. Both of those later events were accompanied by the sack of a city and the dissolution of the pre-existing government, not to mention the horrific crimes that the populace were subjected to...

The situation in the west was a little different. There the empire went out with a whimper and not a bang. Often the last emperor mention is Romulus Augustus who was deposed by Odoacer and allowed to live in retirement. This is all true, but ignores Flavius Nepos who died a few years later who also claimed the title of emperor.

But what was the situation life for the majority of people at this time? It depended a great deal on where you were during this time. Many parts of the empire hardly noticed the "fall" of the empire. It might have been obvious that a lot of the empire was under new management for particularly aware people, and some parts of the empire had suffered rather dramatic collapses.

Britain and Northern France underwent a massive economic collapse in the early 5th century, and the Roman way of life, urban living (by the standards of antiquity), specialized economy (since internal trade within the empire allowed it), public offices, Christianity (in Britain), and Latin writing disappear from large parts of the country side. Britian especially also underwent a rapid shift away from Roman life to a new model, that drew upon Roman, Germanic, and Celtic antecedents. The new societies that arose derived their legitimacy from military conquest (or the myth of it) and paganism, not their connection to Roman legacy. The situation was somewhat different for Wales, but economic collapse still drastically reduced Roman life in this part of Britain too.

In much of the rest of the empire the situation was different. While the long term economic trend was towards contraction, this process took a very long time in places such as Italy, Africa, Iberia, and Mediterranean Gaul. Here there is evidence of continued long term trade and Roman life, such as the Senate continuing into the 600's, local notables continuing their Latin inscriptions, and so on. Indeed in Italy the Ostrogothic Kingdom went to a great deal of trouble to cultivate an image of Romanitas that included maintaining the public works of the Roman period, patronizing court culture, written legal systems, and other affects of Roman life.

So in some parts of the empire, the collapse was relatively rapid and dramatic, combined with invasions, culture change, and new states that did not derive their influence from Rome's legacy. In other parts of the empire, life continued on as normal though with the economy on a downward spin. The guys in charge still spoke Latin, long distance trade continued, and the facade of Roman continuity was vital to the self-image of many of these new realms.

Now if you're referring to the city of Rome specifically its even harder to tell, as our written sources from Italy start to dry up a little bit around this time. The city was of course sacked in 410 by Alaric, but this seems to have done relatively little damage for the long term health of the city. An enormous amount of damage was done to the citizen population and a great deal of wealth was physically removed from the city, but it was not abandoned or anything of the sort following this. Public inscriptions continue, but the population of Rome started to drastically downsize as pieces of the empire started to be removed from the Roman economy. The loss of Africa was particularly devastating as the grain shipments from North Africa were critical to Rome's huge population. With the loss of this vital lifeline, Rome's economy had to de-specialize and there is some evidence of land owning patterns and cultivation patterns in Italy changing with this shift. This meant that there were a lot fewer people in Rome, and the decline was relatively rapid, happening over the course of only a couple lifetimes.

Let me add a few things to @steelcan909's excellent answer.

The date of 476 for the "Fall of the (Western) Roman Empire" is not only completely arbitrary, it was long after-the-fact -- that date was chosen by Edward Gibbon in his Decline and Fall of the Roman Empire (published between 1776 and 1789). Gibbon loved the date for its symmetry: Romulus was the first Roman king, Augustus was the first Emperor, and Romulus Augustulus was the last Emperor. But even he recognized it as somewhat arbitrary. What had been happening for some time in the Western Empire was that prominent Barbarians had become the de facto rulers, but maintained puppet (ethnically) Roman emperors as part of the fiction of rulership. Perhaps the most notable of these was Flavius Stilicho (359-408CE), who was the closest thing to an actual ruler of the Empire from 400-408. (cf. the Codex Theodosianus, Claudian's panegyrics, and Zosimus' Historia Nova for primary sources on him -- I don't have any good secondary ones OTTOMH)

In later years, these puppet Romans were appointed by the (Eastern) Emperor in Constantinople. The main thing that happened in 476 was that Odoacer essentially abandoned this pretense, overthrowing Romulus Augustulus and informing the Eastern Emperor Zeno that Rome would no longer be needing his appointments. This was important for palace politics in Constantinople, but there's no evidence that the average person noticed or cared.

In fact, a better way to understand the transitions in the Western Roman Empire is to rewind a few hundred years and look at the more profound transition between the Early Empire (aka the latter part of Classical Antiquity, say from the time of Caesar through Hadrian in 138CE) and Late Antiquity (starting ca 270 -- if you really want a firm date, Diocletian's accession as Emperor in 284CE is a reasonable one). The "circas" in this emphasize that this wasn't a discrete transition, where one day people woke up and said ZOMG WE ARE IN LATE ANTIQUITY WTF, but rather a continuous shift. This shift, though, was the more meaningful one which people were aware of: everything that happened from ca 276CE to ca 900CE in Western Europe (periods commonly referred to as "Late Antique" and "Christian Dark Ages," although there are good reasons to avoid both) was really a continuous transition from that point.

Perhaps the best thing to read, if you want to get a sense for life in this period and only want to grab one book, is Peter Brown's The Cult of the Saints. While it's notionally about the rise of saint-worship in this period, Brown is one of the foremost scholars of Late Antiquity, and in the process gives a remarkably clear vision of how daily life was changing.

If I had to capture one key aspect of the change, it's in the notion of "Romanitas" -- what it means to be a Roman. In the Classical world, "Romanitas" was tied up very closely with concepts of being a citizen of a powerful, expanding empire. Roman relations with other peoples were primarily ones of conquest, with a relationship approaching that of (tense) equality happening only with the very largest other empires (such as Persia). New wealth and new riches flowed into the empire continuously, and new people became part of the Empire -- first as captives and slaves, later as subordinates, ultimately becoming effectively as central to the Empire as Romans themselves.

If you lived in the wider Roman Empire during this period, you would have seen local potentates become part of a system of clients and patrons, intertwining them with central power. They would have been able to tap into Imperial resources when local problems became an issue, and they would in turn have been sending taxes (and taxing you) to send back to the treasury. Depending on your own affiliations and where you were, you might have been opposed to Roman rule (e.g. in Roman Palestine, a notorious hotbed of insurrection -- the People's Front of Judea is a pretty damned apt joke), might have considered it an inevitable annoyance, or might have actively profited from it.

This idea worked really well up through roughly the reign of Hadrian, which is where Rome reached its effective logistical boundaries. Campaigning in northern Britain was simply not worth the effort, and expanding in most other directions was even worse. Then the troubles of the transitional period (ca 138-270) really kicked in: a sequence of droughts, plagues, famines, civil wars, invasions, and so on. Crucial structural weaknesses in the system -- the lack of systematic tax collection and long-term financial planning, the haphazard method of succession planning, and so on -- turned into major liabilities. During this period, central Roman rule over the bulk of Europe became somewhat more theoretical, with local leaders as likely to rise up and challenge Roman authority as Roman leaders were to try to (re)conquer localities. Between 235 and 284CE, there were twenty-six emperors, up to seven at a time (which is to say, raising armies and warring against each other).

If you were living in the Imperial countryside during this period, you would have seen a massive retraction of Roman power; anyone who depended on it for their own influence was SOL. Major disasters (famine, drought, plague, invasion, etc) became more common, and help was generally not forthcoming from central sources.

Starting gradually in the mid-3rd century, a new notion of Romanitas started to emerge. In this new world, Romans couldn't see themselves as the obvious superiors of other peoples, with the only thing preventing Rome from conquering them was how low that country was on Rome's TODO list; they had to forge relationships on a much more equal basis with a wide range of people. What turned out to be the new unifying theme was Christianity.

Important clarification: I am not talking about the actual religious content of Christianity here, which in fact was in tremendous flux during this period. Rather, the key idea was a social one: "You're a Goth and I'm a Roman! But oh, we're both Christians, so we have a common language and can deal with one another." Christianity replaced the idea of "being a Roman citizen" as the common language which could be used to open dialogue between people.

[part 1/2]