This is an immense question because feudalism was a very complicated system. You might want to follow this link to view a very detailed discussion of feudalism.
In general, feudalism was a system that first came into existence in Europe after the "fall" of the Roman Empire. It was, in theory, a hierarchical system in which people at each level of society owed allegiance to people above them, all the way up to the monarch who was at the top of the system. Each person owed allegiance to their lord. Their lord, in turn, was supposed to take care of them and to ensure that they were treated fairly and with justice.
As an example of this, each king was the lord of various nobles. Those nobles were supposed to obey the king and provide him with tribute or taxes and with people to fight for the king when necessary. In return, the king was supposed to provide justice and safety for the nobility.
In practice, feudalism was not very centralized. This was not a system in which kings were really able to tell their vassals (the nobility) what to do in all instances. Instead, feudalism was a system that came out of the weakness of kings. They could not conquer all the nobility so they had to let them maintain power in the feudal system.
The feudal system lasted in some forms in Europe up until around the 1500s.