Just to save time for others who would be able to respond in detail: do you mean the collapse of the Western Roman Empire in the 4th century, or that of the Eastern Roman (Byzantine) Empire in the 15th century?

Note: I'm not a Byzantist; I'm a student studying church history. However, I haven't seen a response to OP yet, and I feel pretty strongly about Byzantium, so I'll write a response. However, if a Byzantist does respond, please take their response over mine, as they no doubt have significantly more experience.

The Roman Empire continued on in a reduced form out of Constantinople, consisting of a heartland of Greece and Anatolia. This medieval, Christian empire is often called the Byzantine Empire, or Byzantium, in reference to Constantinople's old name, in order to distinguish it from the pagan, Latin-speaking empire of classical antiquity. The fall of the city of Rome wasn't the end of the Empire, and, in fact, it didn't really fall in a clean and easy sense - its conqueror was referred to as the Patrician of Italy, and was seen as subservient to the Emperor.

While the history of the empire is fascinating, it's incredibly long - the last bastion of the Roman state, Constantinople itself, didn't fall until 1453, just as the Ottoman Turks were beginning their impressive expansion over the eastern Mediterranean. Because of its length, it's simply impractical to write that much about it. However, I can give you a brief comparison of how it changed from the image of classical Rome, and talk about some notable figures and events.

For much of its history, Byzantium was Greek speaking. Supposedly, Emperor Justinian I, whose name I took on Reddit, was the last native speaker of Latin to be emperor. Of course, the empire was ruled from Constantinople, not Rome. The religious policy evolved a bit, too - whereas classical pagan Rome sought stability by recognizing numerous faiths, Christian Byzantium found strength in religious unity. The emperor worked closely with the clergy to organize meetings like the Ecumenical Councils; the clergy were concerned about dealing with heresy and saving souls, while the emperor had the additional concern of keeping the empire united.

The history of Byzantium is fascinating, with numerous colorful figures and dramatic events. To name just a few and give an outline of its history: •Constantine I, famously the first Christian emperor of Rome. He is sometimes considered the first Byzantine, since the distinction is a bit fuzzy. He turned the Greek town of Byzantium into a New Rome, Constantinople, and called the first Ecumenical Council in Nicaea.
•Justinian I, my personal favorite. Justinian was involved somewhat in peacetime affairs, building the great Hagia Sophia in Constantinople, overseeing the Miaphysite controversy within the Church, and trying to deal with a devastating plague. However, he was a warrior emperor. During his reign, he expanded the empire, at that point consisting of the Balkans, Anatolia, the Levant, and Egypt, by reconquering Italy, all of North Africa, and a bit of Spain! •The iconoclasm controversy, an issue far too complicated to explain in a little blurb like this. In short, the issue arose of whether using religious iconography counted as idolatry or not. This debate shook the Church, and probably contributed a bit to the Great Schism of 1054.
•The Crusades. Byzantium was a large part of the First Crusade's very existence; western Christians were, at least theoretically, coming to the empire's defense against the Seljuk Turks. However, the Crusades were a disaster for Byzantium; in 1204, the Fourth Crusade ended with a Latin Christian sack of Constantinople, leading to the empire disappearing for a few decades.
•The fall of Constantinople itself, in the reign of Constantine XI, in 1453. This had huge implications on the Christian world - refugees from the Empire may have contributed to the renewed interest in the classics that characterized the Renaissance, and Turkish control over the Bosporus led Europeans to look for an alternative route for the spice trade, which led to the discovery of the New World.

This is just a tiny bit of Byzantine history, though - there's so much more to it, like its beautiful culture, colorful figures, and distinctive focus on religion. Fortunately, there is no shortage of primary sources, and historical study of Byzantium seems to be growing more common, so there is plenty for you to read on the subject.