That's a good question, but quite a loaded one. I'll have to begin with a big caveat r.e. the term feudalism/ feudal system. The idea of feudalism that most people get from school textbooks as this perfect pyramid of power in which the king owns everything and gives out blocks of land to the barons who in return perform administrative duties for him and provide military service, who in turn give out smaller parcels of lands to the knights who provide them with 40 days military service for them a year, is an invention of legal historians in the 16th - 18th centuries.

In reality, there was no hard-and-fast indissoluble link between landholding and military service, except for in the minds of lawyers in the service of kings and princes, and things like military service were subject to negotiation. For example, while by the 1270s the Capetian kings of France had managed to establish the right to 40 days military service a year for the barons and knights of the royal domain (the lands under the direct control of the French crown), when King Philip III went campaigning against the rebel count of Foix (a region of South-Western France right by the modern-day border with Spain) in 1272, his knights told him that they were going to go home before they'd even arrived there for their 40 days was already up. By the end of the 13th century, the French kings had managed reached a new settlement with the French nobility - all holders of noble fiefs had to either serve in person when summoned on campaign for an indefinite period of time in return for wages or pay a tax to the king's bailiff in their region to exempt themselves.

For further reading on the critique of the concept of feudalism Elizabeth Brown's "The tyranny of a construct" (1974) and Susan Reynolds' "Fiefs and Vassals" (1994), but for a defense of it see the first chapter of Chris Wickham's "Medieval Europe" (2015). Wickham basically argues that while obviously the traditional legalistic approach to feudalism is no longer tenable, at the same time its still helpful to describe a society based around hierarchies of landowners in which political and military organisation is loosely based around personal vertical ties between these landowners.

So how did this kind of system emerge. Lets start with the Roman Empire. In the 4th century AD Western Roman Empire the provinces were governed by salaried career officials based in cities and the empire's frontiers were protected by a standing army which may have been as many as 400,000 strong, and the whole system was supported by an annual land tax done in a stable, state-provided gold coinage, the burden of which mostly fell on the peasantry who were tied down to their lands by imperial law.

This system doesn't immediately collapse with the barbarian invasions, but by the 7th century the old Roman bureaucracy, standing armies and any kind of systematic taxation of the land had disappeared from all of Western Europe save for a few enclaves in Italy controlled by the Eastern Roman (Byzantine) Empire, and all the old Roman cities in Western Europe had drastically shrunk in size as a result not just of war but also of plague and global cooling (Rome itself may have had as many as a million people in the 1st century AD and about 600,000 in the 4th century, but by the 8th century its population had shrunk to as little as 30,000). This is a period which used to be called "the Dark Ages", and some historians used to think that Western Europe in the 6th - 8th centuries was a kind of Mad Max scenario in which Germanic warlord-kings with their posses of tribal warriors just went around plundering and terrorizing people while maintaining their ties between each other through gift-giving, feasting and heavy-drinking - there's quite a lot in the writings of the 6th century Gallo-Roman historian Gregory of Tours, the 8th century Northumbrian (Northern English) historian Bede and the Old English epic poem (of uncertain date) "Beowulf" which appears to support this picture, But historians since the 1970s have stressed that a lot of elements of late Roman government, no less important than those which disappeared, survived.

Come the time of Charlemagne (748 - 814), the Frankish king who unified all of modern day France, the Low Countries,, Germany, Italy, Switzerland, Austria and Catalonia and was crowned Roman Emperor in the West by the pope in 800, we still have a kind of public state. Under Charlemagne and his successors, the administration of the provinces was done by officials known as counts, normally from aristocratic backgrounds, who would be granted benefices (blocks of land that could be revoked at any time, and would normally revert to the king upon the death of the recipient of the benefice) and shares in fines and market tolls in return for running the public law courts, assembling armies and stuff. The military system was a bit complicated. Theoretically military service was the obligation of all free men from the most favoured nobles at the king's court down to free peasants. In practice it was normally the king/ emperor, the counts and their respective household retinues of professional warriors that most consistently served on campaigns (which early in Charlemagne's reign were quite a long way away from home), though an imperial edict of 808 issued in response to the mounting Viking threat required that all able-bodied free men aged 18 - 60 who owned 5 manses (a unit of land) or more had to serve in the king's army in person, and those who owned less had to club together to send one of them to serve.

By the late 9th century, however, as central authority in the Carolingian Empire declined due to dynastic infighting and the continued menace of the Vikings, the counts were able to make their positions and beneficed landholdings hereditary and a shift to cavalry warfare in order to stop the fast-moving Vikings meant that military service became more and more elitist, limited to nobles and their followers, while the free peasantry were simply obliged to do construction work on local forts and bridges.

Still, the counts maintained a kind of public government in the various regions of West Francia (modern-day France), maintaining the public law courts and creating their own beneficied officials (the vicarii or viscounts), until the late 10th and early 11th centuries when a process that historians now call the "feudal revolution" happened. Basically this involved castle-holding lords (castellans) and their retinues of armed warriors referred to as milites/ caballarii in the Latin charters (or as knights in vernacular) seizing power at the expense of the counts in the localities, privatising the judicial system, building castles etc. Some counts and dukes managed to maintain some degree of effective central control, the counts of Flanders, Anjou and Barcelona and the dukes of Normandy being good examples, but they maintained their control over these increasingly powerful and insubordinate lesser landowners through ritualised personal ties of mutual obligation i.e. I will protect your lands and provide justice for you at my count's court when you quarrel with your neighbours, provided you provide me with military service, give me what advice I want etc and you can leave me if I try to kill you, trick you, shag your wife etc. These personal ties had their roots in Carolingian era legislation going back to the 9th century, but they acquired a new prominence in this period. I'd recommend having a look at this "agreeement between Count William V and Hugh de Lusignan" from the 1020s, as its very illuminating about the kind of things we're describing - here's a link to a translation of it from Fordham University Medieval Sourcebook https://sourcebooks.fordham.edu/source/agreement.asp

You've probably noticed my answer has mainly focused on modern-day France and neighbouring regions, because that's the area I know best for this period. Its fair to say that Germany, Christian Spain, England and Italy took very different paths of development and developed "feudal systems" quite different to the French one.

For further reading:

Chris Wickham "The Inheritance of Rome: A history of Europe from 400 - 1000" (2008) - the best introduction to early medieval Europe currently available

Georges Duby "the Chivalrous Society" (1977) - a great collection of quite technical yet still very readable essays by one of the most influential French medievalists of the 20th century, who essentially invented the whole concept of the "Feudal Revolution"

Thomas Bisson "The feudal revolution" (Past and Present 1994), a very controversial and extreme yet influential take

I'm not sure what to recommend for Germany and Spain, but for England I'd suggest reading James Campbell's "The Anglo-Saxon State" (2002), for the apparent lack of the classic features of feudalism in pre-1066 England, and David Carpenter's "the Struggle for mastery: a history of Britain 1066 - 1284" (2005) for how they were grafted on to a strong, and indeed strengthening, monarchy post-1066.

For a lively introduction to a lot of the problems regarding this field, do also check out Dr Jonathan Jarrett's excellent blog https://tenthmedieval.wordpress.com/

Regarding the orders of knighthood question, that deserves a whole other thread in itself. assuming that you meant orders of knighthood rather than knights as a social order/ class, then the first exclusive orders of Knighthood were the knights templar (founded jointly in 1118 - 1127 by the knight Hugh de Payens and the Cistercian abbot St Bernard of Clairvaux) and the Hospitallers (founded by the Blessed Gerard in 1099 - 1113), formed in the early 12th century in the aftermath of the First Crusade. Basically, even after the First Crusade, professional warriors who spent most of their careers fighting fellow Christian warriors and the lay aristocratic way of life (in which feasting, heavy-drinking, dressing lavishly and loving luxury etc figured quite prominently) seemed quite opposed to christian teaching - this was at a time when the papacy was attempting nothing short of a total moral reformation of both the clergy and the whole of Christian society. So military orders were established in which knights could live like monks under a strict rule and taking monastic style vows of poverty, humility, chastity etc, while at the same time being able to bear arms and fight, but almost entirely for the defence of fellow Christian pilgrims against Muslims in the Holy Land.

By the 14th century, however, we start to get new secular orders of chivalry, founded by monarchs who wanted to create stronger ties of camraderie between them and their nobles, often consciously emulating King Arthur and the Knights of the Round Table in the Arthurian Romances or Charlemagne and his 12 Peers in the collection of romances known as the Matter of France, as well as patriotic feeling. The still-extant Order of the Garter, founded in 1348 by Edward III, whose patron saint was England's recently declared national saint St George, is a case in point. Another, more forgotten example is the Order of the Star, founded in 1351 by King John II of France to reform French knighthood in the wake of the disasters of the Hundred Years' War. John II got the veteran knight Geoffroi de Charny to write a "Book of chivalry" (a translation of it is available from university of Pennsylvania press, with an introduction by leading historian of chivalry Richard Kaeuper) that outlines all the desired disciplinary and moral reforms of French knighthood, as well as providing brilliant insight into how a practicing knight, widely respected for his ideal chivalric conduct by his contemporaries including those on the enemy side, saw his own vocation. That order didn't last very long however - most of its members were wiped out by the battle of Poitiers in 1356, including Geoffroi de Charny who died poetically bearing the Oriflamme (the royal standard).

For further reading, I'd recommend "the New Knighthood: a history of the order of the Temple" by Malcolm Barber (2012) and"Holy Warriors" (2014) and "Medieval Chivalry" (2016), both by Richard Kaeuper.