The early part of this period was characterized by the fragmentation of political power, a decrease in population, and the decline of cities in what had been the western half of the Empire. (Note: The eastern half would continue to exist until 1453.) As the government fell apart, public institutions and communication across what had been the Empire followed. Power, land, and learning all became privatized. The Church tried to maintain some of the political structure, but was generally too weak to fill the void left by the Empire. Powerful men set themselves as local lords answerable to no central power and consequently spent a certain amount of time fighting over land. Lack of communication and the need to focus on survival meant that education and learning were primarily restricted to monasteries.
Things did not stay this fragmented though. Barons and kings began collecting more land and other lords under their power. The biggest example here is Charlemagne.
[Charlemagne and Pope Adrian I]
[Painted By Antoine Vérard c. 1493, Source: Wikimedia Commons]
[Frankish Empire 481 to 814]
[Sémhur, Source: Wikimedia Commons]