This is what I study in the real world, and partially what my thesis is about, so there’s a lot to say about it! Hopefully this answer doesn’t go on and on forever…I think I've written most of this in previous answers, so here they are, all collected in one spot.

The Frankish crusaders did try to import a European-style “feudal system” in some ways, but in other ways they also adopted local customs.

The most important thing to remember is that Jerusalem was much more diverse than France. It was more like Spain or Sicily, so they didn’t create a completely unknown type of society, but it wasn’t anything like France or England. In Jerusalem there were Greek Orthodox, Armenian, Coptic, Nestorian, Georgian, Ethiopian, Syriac, and Maronite Christians. There were also Jews and a small number of Samaritans. There were Sunni and Shi’i Muslims, offshoots of the Shi’i like the Druze and the Nizaris. Some Muslims and Christians were Arabs, some were Turks or Kurds. When the First Crusade arrived, there was a Sunni Abbasid caliph in Baghdad, but he was mostly just a figurehead, and political power was held by the Seljuk sultan. Several other Seljuk emirs and atabegs ruled the cities of Syria, somewhat independently.

In 1099, Jerusalem was controlled by Shi’i Fatimid Egypt. The Sejuks had captured Jerusalem from them in 1070, but the Fatimids took it back in 1098 while the Seljuks were distracted by the crusade, then the Fatimids lost it again to the crusaders in 1099.

We don’t really know how many people lived in the crusader states, but one estimate (by Josiah Russell) suggests that all of Syria had about 2.3 million people at the time of the crusades, living in eleven thousand villages. Three hundred and sixty thousand of them lived in the Kingdom of Jerusalem, and two hundred and fifty thousand of those lived in rural villages. (Probably, anyway - we really have no idea about actual numbers.)

According to the Persian traveller Naser-e Khosraw, who visited around 1050, the population of Jerusalem was about twenty thousand people, and the city was always full of Christian, Muslim, and Jewish pilgrims. Ibn al-Arabi, who visited Jerusalem only a few years before the crusaders arrived, mentions Sunnis living in Jerusalem, but says the cities along the coast and elsewhere in Fatimid Palestine were majority Shi’i. Christians and Muslims had lived together in Jerusalem until 1063, when they were segregated into separate quarters. The Christian community controlled the Church of the Holy Sepulchre, and, outside the city, the town of Bethlehem. The Holy Sepulchre had been destroyed by the Fatimids in 1009, but was rebuilt starting in the 1050s (and then the crusaders later modified it into its current structure in the 1150s)

Under the Muslims, Christians and Jews were dhimmis and paid a specific tax, the jizya. The cities were sometimes governed by a qadi (an Islamic judge) or a ra’is (Arabic for “head” or “mayor”), or more directly by a Seljuk or Fatimid governor. In rural areas there was also a semi-feudal system that the crusaders found familiar and easy to adopt: land was granted as an iqta' and revenue was collected by the muqta', who could rule the iqta' as an independent fief.

So, that was the situation that the crusaders found when they arrived in 1099. They were now the rulers of this complex, ancient society, and they didn’t really disrupt it too much. They left the agricultural system in place, and the tax system - but now Muslims had to pay the jizya to the crusaders. There were never enough Franks to displace all of the Muslim or eastern Christian leaders, so in all of the towns and villages under crusader rule, there were still qadis or ra’is. The Franks found it easy to adjust to the iqta’ system, so they simply interpreted an iqta’ as a fief and a muqta’ as the fief holder, who owed them taxes and services, just like a fief back in France.

They had their own system of fiefs on top of that - in Jerusalem, the king had four major vassals (the Prince of Galilee, the Count of Jaffa, the Lord of Sidon, and the Lord of Oultrejordain), and each of those fiefs had smaller fiefs of their own, and these were all governed by Frankish lords, but the villages and towns and farms within those fiefs were left to the Muslims or eastern Christians who already lived there.

Or at least, this is what we're told by one of the crusader lords, John of Ibelin, who was the Count of Jaffa. According to him the County of Jaffa was the most important one. Of course he would say that! It also looks like John of Ibelin was painting an incredibly idealized picture of how “feudalism” was supposed to work in Jerusalem, and not necessarily how it actually worked in everyday practice. Historians in the past usually took him at his word, so Jerusalem has been described as having some sort of “perfect” form of feudalism, but we don’t think of it that way now. “Feudalism” itself is a pretty questionable concept these days (there is a lot of info in the AskHistorians FAQ about that so check that out as well). But in Jerusalem, like in France or England, theoretically the king and his vassals collected taxes from the people who lived in their territories, and those people also owed them military service or some other kind of service. John of Ibelin even lists how many knights each fief owed to the king (he himself owed 100 knights), but again, that is probably an idealized picture.

The biggest difference in Jerusalem was that the fief holders didn’t always live in their fiefs. They tended to live in Jerusalem or Acre or one of the other major cities, and they collected income from their fiefs, but these were “money-fiefs” for them. There was a lot of agricultural land in Jerusalem, but it was also much smaller and much more urbanized than, say, the north of France were a lot of crusaders came from.

The crusaders were “Franks” so we tend to think of them as “French”, but there were lots of people from all over Europe: the chronicler Fulcher of Chartres, who participated in the crusade and spent the rest of his life in Jerusalem, lists French, Flemings, Frisians, Swiss, Germans, English, Scots, Italians, and Bretons, among probably many others. Fulcher famously wrote that:

“…we who were Occidentals have now become Orientals. He who was a Roman or a Frank has in this land been made into a Galilean or a Palestinian. He who was of Rheims or Chartres has now become a citizen of Tyre or Antioch. We have already forgotten the places of our birth; already these are unknown to many of us or not mentioned any more. Some already possess homes or households by inheritance. Some have taken wives not only of their own people but Syrians or Armenians or even Saracens who have obtained the grace of baptism...People use the eloquence and idioms of diverse languages in conversing back and forth. Words of different languages have become common property known to each nationality, and mutual faith unites those who are ignorant of their descent...He who was born a stranger is now as one born here; he who was born an alien has become as a native.” (Fulcher of Chartres, pg. 271)

They all tended to congregate together in the cities, or in rural areas where fellow (eastern) Christians already lived. They didn’t really interact with the Muslims, but that’s probably how it was before the crusaders arrived as well. Everybody kept to their own villages and didn’t mix, so the crusaders did the same. The Franks also established new villages, which is sometimes considered an early form of colonialism. The most famous of these new settlements is probably Bethgibelin. The settlers at Bethgibelin came from:

"...Auvergne, Gascony, Flanders, Lombardy and Catalonia. Generally, the largest number of European settlers...were from the central, southern and western parts of France, and a few also from northern Spain and regions in Italy. In Bethgibelin the other settlers were from nearby Latin villages..." (Nader, pg. 94)

The towns and cities, especially along the coast (Acre, Beirut, Tyre, etc) were also full of Italian merchants and settlers. The Italians were kind of a state-within-a-state, since they governed their own neighbourhoods semi-independently of the Frankish lords. They were responsible for overseas trade with Europe - we know basically everything that was imported and exported from the crusader states by Italian and Muslim merchants - they sold dates, onions, sugar, cinnamon, cardamom, nutmeg, linen, shoes, and hundreds of other products! Sugar was especially important - the Italians also ran sugar plantations and this was probably the first time Europe had seen sugar from cane plants. (They also traded “marshmallow” and “liquorice”, but those were medicinal plants, and not, as I like to imagine, the modern candy.)