The feudal system was based on a
bond of loyalty between the king and his
vassals. In exchange for land (a
fief), which was just about the only
fortune in those days, they pledged loyalty to their king. Since these vassals in their turn could give away land to others (or each other) in exchange of their
loyalty, these bonds of loyalty quickly became rather complicated. In some cases, someone could be a vassal to two enemies and would, in theory have to be loyal to both of them and supply them both with soldiers in their war against each other. In some cases, a vassal could become more powerful than the king he
served if he had enough land and soldiers. He would still have to be loyal to his king though. (And then there's that thing with the
king of England being vassal to the
king of France.)
The feudal system emerged when the large European kingdoms had problems with raiding peoples such as the Vikings and Magyars. When the raiders plundered one part of the country and the king was in another part it was very hard for him launch a counter-attack in time, considering the bad communications of that time. In order to deal with this problem, the king would give a piece of land to a person who, in exchange pledged to defend it and its inhabitants and to be loyal to him. This system was sometimes also used to deal with troublemakers by bribing them, such as in the case of Normandy where the king gave the Vikings a large piece of land in order to gain their loyalty and to stop their plundering.
The lords lived in castles in which they and the people who lived around them protected themselves during enemy attacks. The peasants who cultivated the lands where serfs and where not allowed to move without being granted to do so by their lord. They where also required to pay tax to both their lord and to the local priest (tithe).
The feudal system resulted in that people moved much less than earlier. Trade almost ceased outside the cities. Since there wasn't any schools or similar things, all "research" took place inside the monasteries.
Why the feudal system fell
During the end of the medieval ages, large changes took place in farming that made it possible for fewer persons to grow more food on a given piece of land. One of these changes was a change in climate during this period that made it warmer and thus it became easier to grow crops. Thanks to the invention of the collar, the horse was also beginning to be used in the fields. Earlier, oxen who where much slower than horses had been used. The problem with horses was that they where very choosy about what they ate compared to the oxen who ate pretty much anything. Another thing that increased the ability to grow crops more efficiently was the double and triple shift. In the double shift system half of the fields where laid fallow while the other part was cultivated. This way the field wasn't leached out. In the triple shift system, the double shift system was made more efficient by letting one third of the field lay fallow instead only one half.
All these changes meant that the work on the farms could be performed by a smaller number of people, leading to more people moving in to the cities, making them grow in size. The growth of the cities meant increasing trade which made the merchants rich. This led to a transition to a currency-based economy and one no longer had to own land in order to be rich. This was one of the reasons for the fall of feudal system in Europe. (Another major reason for the fall of the feudal system was the plague, but I don't know very much about that.)