led to power struggles, which led to a weakened army, and was compounded by the rising strength of the clans in Britain.  Some soldiers just left their western posts, however some of the soldiers that stayed in Roman Britain actually became quite well off.  There evidence that suggests that the men of the legions that stayed became what later becomes known as the first nobles, along with the strongest of the Britain clans.  Ultimately this strange relationship gave way to a sort of mixing of the power.  Those of the Roman military still had might and for the most part superior fighting skills, those of the clans had territorial and a patriotic autonomy. Politically the leaders of both groups served a purpose for the rest of the people...the people needed protection and were willing to pledge their support primarily through a new economic relationship with these new leaders. This new relationship was based upon a political and economic exchange between those who gained power and retained power and the masses that surrounded them, this exchange was not dominated by the Roman empire but by smaller, more self sufficient, and decentralized societies prompting the early stages of individual kingdoms that would come to dominate western Europe.

With the fall of Rome, an empire came to an end. In other words, all the benefits of an empire were over. This would include, good functioning communication, broad infrastructure, such as roads, a centralized taxation system and even safety to some extent. What came  to pass instead was a number of groups, such as the Franks, Goths, etc. These were independent groups that often fought each other. Eventually all of this lead to feudalism and more independent kingdoms.

This is a pretty big question, and the answer is not the same for every time and place after the fall of Rome.  But in general, the ideas of decentralization and self sufficiency go a long way to describing what happened.

After the fall of Rome, Europe split up into all sorts of little political entities.  Where Rome had once ruled a large part of Europe, it now came to be ruled by "nobles" who controlled as much land as they could.

Self-sufficiency became more important after the fall of Rome.  Now that Rome did not control everything, it became very dangerous to travel and, therefore, to trade.  There were too many people who might steal a trader's goods and money.  So there came to be much less trade and areas needed to be self-sufficient.

The Fall of Rome was not one immediate collapse of the Empire, but a progressive weakening of Roman territory and government, culminating in 476 CE with the deposition of the Roman Emperor. A number of reasons for the Fall of Rome have been proposed- including everything from lead poisoning to crippling debt. In addition to these contributing factors, repeated invasions by Germanic tribes dealt heavy blows to the spread and power of the Empire. After losing significant amounts of territory in the North to such Germanic tribes, and the Sack of Rome in 410 CE, the warlord Odoacer unseated Emperor Romulus Augustus and became the first King of Italy. 

As the power of the Roman Empire declined, so did its infrastructure and connectivity. As officials (governors, military) stopped visiting the far reaches of the Empire, public works like roads well into disrepair. It is sometimes said that Rome "collapsed from the outside inward." It became increasingly difficult for the Empire to be governed over such a large space, and the retreat of Roman power left a gap. During the Middle Ages, this gap in power was filled by local kings and warlords rather than officials of the centralized Roman government.

Under the Roman Empire, all territories fell under the name and authority of Rome. During and after the Fall, far more localized political bodies sprang up. Kingdoms developed, and here, the "biggest and baddest" were in charge. Western Europe after the Fall saw a high degree of competition for power, and whoever had the most cunning, wealth, or military strength could easily make themselves King of a region. These local warlord kings often fought with one another, acquiring and losing territory so quickly that it was sometimes difficult to know where one's loyalty lay. (This was one of the problems which eventually lead to the downfall of the feudal system.) In this kind of political environment, the goal was to acquire and rule over as much land as possible while maintaining the trust or obeisance of the people.

Some of these kingdoms survived for many generations, through familial dynasties of rule. One great example is the Kingdom of Francia (or the Frankish Kingdom) which produced rulers like Clovis I and Charlemagne. The Frankish Kingdom was so great under Charlemagne's rule that he is often called the Father of Modern Europe.

In short, after the Fall of Rome in the West and the failure of centralized governance, political power was returned to local warlords who competed with each other for territory.