Feudalism in Europe became established as a reaction to the crumbling of the Roman Empire.  When trade flourished under the Roman Republic, most of the wealth created was by trade.  When the Republic became an Empire, trade still existed, but eventually through government restrictions and taxation, trade slowed, and finally, the Emperors, by destabilizing the currency, caused trade across the Roman world to shut down. Since wealth could no longer be created through trade, the merchant, or trading class, disappeared, and the social breakdown throughout former Roman provinces accelerated. The only source of wealth creation left was in the land, or in agriculture.   Ironically, the focus on agriculture did not increase agricultural output or wealth; as there were no longer any markets to trade with, each "agricultural centre" became its own economic entity.  The Roman aristocracy, tired of the intrigues of Roman government, retreated to their country estates, which became self-sufficient, and created the well-known feudal model of landowner and serf which lasted for nearly a millennia.

Feudal groups were defined as mutually reciprocal obligations for legal and military rights and privileges of land holdings between the three powerful classes of (1) kings/lords; (2) vassals, pledged to military service; and (3) fiefs, pledged to service that may be other than military service, though all three classes are defined as warrior nobility. The feudal agreement solved the social problem of land ownership and military collaboration.

Peasants were not part of the definition of feudalism since they were seen as degraded humanity, worthy only of providing a labor force for farming and military purposes, unworthy of notice, and incapable of making an honorable social contribution. Historians extended the concept of feudalism to include the peasant work force under the umbrella term "feudal society." Sir Arthur Conan Doyle (of Sherlock Holmes fame) has an excellent novel set in the feudal era called White Company that illuminates a social problems caused by the feudal system: neglect of degraded peasantry resulting in brutal treatment and starvation.

Whilst I agree with #2 in the stability that was created through feudalism, I would also like to point out that it also created massive inequality as serfs became dependent upon Lords for everything. Because there was very little restrictions placed on lords, it meant that more unscrupulous individuals were free to exploit their position of power and make the lives of their serfs very miserable indeed.

The most interesting thing I discovered about feudalism is that it wasn't a perfect "pyramid" with the lord or king at the top, the vassals & knights in the middle, and the serfs at the bottom. Often knights and vassals pledged themselves to more than one lord and serfs served several masters simultaneously. While it might not have been the perfect governing system, it did serve a purpose in that the lords (or vassals) provided a home, a job, and protection to the masses; in turn, loyalty, service, and protection were provided by the masses to their lords. 

The major problem that feudalism caused was instability.  Feudalism created a situation in which there were various lords always competing among themselves for power.  Often, a lord or group of lords would feel powerful enough to compete with the monarch for power.  Therefore, there was constant conflict during the time that feudalism was the major organizing principle in Europe.  (A similar thing happened in Japan as well.)

This is a huge question, and there will be lots of responses addressing different aspects.

I'll start the responses by pointing out that the feudal arrangement provided protection and stability for the serfs living in western European societies. Within the walls of the castle was an escape from danger. Outside the castle walls, there was work on the lord's land and a place to live in return for the labor provided.

What societal needs and trends led to feudalism's development, and what problems did it create and solve in Western Europe?

Feudalism resulted from the disintegration of the Roman Empire in the West, and lasted from the Fall of Rome until the Renaissance. As the Roman aristocracy throughout the Empire abandoned the principal cities (Rome began a long period of depopulation staring around the year 500) they reestablished themselves on country estates.  As economies worsened throughout the West, to the point of almost no trade occurring, each estate began to be self-sufficient -- because trade had disappeared, the only value left was in the land. Since someone had to produce on the land, and someone had to secure the land, two offices of lord and serf were established.  The last unifying force in the West, the Roman Church, provided the the last office of priests and bishops to round out the categories of Feudal society.

One of the political causes of feudalism--the agreement between nobles of the warrior class that disregarded peasants--was invasion from other, competing powers. When kings, vassals, and fiefs were all united by service agreements with each other (kings granted lands to vassals and fief, thus making them empowered land owners, in exchange for military and other governmental services), they could present a united successful combat force to invaders. Taxation became important in developing a feudal community comprising people below the noble class level who needed help paying their taxes and besought the help of the manorial vassal or fief lord who would pay the tax in exchange for servitude that was usually in the form of labor on the land. This taxation/labor exchange effectively reduced the middle class.

http://www.newadvent.org/cathen/06058c.htm

I agree that feudalism developed, in part, because the people needed a way to police themselves.  The feudal society developed a code of conduct that people stuck to with strict rigidity.  It helped the society develop and laid the foundation for later laws and societal structure.  Of course, placing the power in the hands of few does tend to lead to corruption and inequality.  The peasants had no say in the ruling of the lands they worked and lived on.  The Lords and upper class held all the power.  Eventually, this inequality and the anger it generated would lead to the downfall of the feudal society.

Well, one of the biggest central problems is of course the massive inequality that was created and the difficulties that there were in ensuring equal care and protection of serfs. The system enabled some lords to become incredibly rapacioius and abuse their responsibilities towards their charges, rather than protect them and ensure their wellbeing.

Modern historians don't really like the term "feudalism" all that much, they prefer "lordship" instead, which was equally reciprocal. Medieval people didn't use the term, or really conceive of it as we do. That said, whatever you call it, historians emphasize the mutual obligations that fulfill both societal and social needs. The institutions that we call feudalism provided for property protection, legal services, revenue raising, and even in many cases, religion, since many bishops were lords as well. All of these services came at the cost of either military service or, more often, payment in kind. The important thing to understand (and maybe a "problem" under the terms of the original question) is that throughout most of the medieval period, the system actually tended to weaken most kings, who were often far less powerful than the lords who supposedly owed them fealty. The concept of "country" or "state" as we would imagine it today was basically antithetical to this system, even though some particularly powerful, ruthless, and charismatic monarchs (e.g. Philip Augustus of France, Henry II and III of England) began to transform their demesnes into what we'd recognize today as modern bureaucratic states.

Feudalism was developed because there was a need for people to fight for the kings as well as the need for people to act as the keepers of law and order in various parts of kingdoms.  This was a time when economies were not strong enough to simply have a centralized standing army to fight for the king.  Kings also did not have enough power or any really effective means of keeping law and order in every part of their country.  Therefore, feudalism developed.