When the Franks came to Outremer, they inherited a subject population which was mostly Christian - though not at all Catholic - with a huge minority of Muslims, along with many Jews, druze, and others. They took the existing governance system of the Muslims and overlaid it with their own feudal concepts. The result was a strange brew indeed!
Non-Christians were subject to their own laws when dealing with each other, and their own leaders dealt with them - rabbinical tribunals for Jews, and qadi for Muslims. They had to pay a tax - this was inherited from the previous Muslim government, and called the jiziya - dress in ways differentiated from Christians, and could not witness in cases involving Christians except for testifying as to a Frank's age and descent, or to testify involving boundaries. On the other hand, they were free to practice their religions.
Non-Catholic Christians were equal, but not as equal as Catholics. They had their own religiously based courts, and their word in testimony was not equal to that of a Catholic. This varied from place to place, and changed over time. By the time Outremer is set, this distinction had mostly eroded, along with the ability of non-Christians to testify, in most Frankish nations.
Orthodox Syrian Christians had their own courts, called the Cours des Syriens, in which they were judged according to their own customs. These courts ruled on minor to moderate matters, and handed up to the higher courts more difficult matters. There were also market courts, called Cours de la Fondes, which had jurisdiction over commercial matters involving multiple ethicities. Cases "of Blood" - that is murder, treason, and theft - were heard by the higher courts. Minor cases involving Christians were heard in the Cours de la Fondes, or outside the cities, in the court of the local feudal lord. The Fonde administered the markets and souks, including taxes, weights, and measures.
The first higher court was the Cour Des Bourgeois, which dealt with most cases in cities - the Bourgeois were citizens of a city, and not covered by feudal law. These courts covered all people of the cities, no matter their religion or ethnicity. Each city had its own laws, the Assises des Bourgeois, which could differ markedly from city to city. The highest court was the Cour de Haut, the High Court, the court of the ruler. In Antioch and Edessa, this was the Prince, in Tripoli, the Count, and in Acre, the Duke. This court tried all cases handed up from the lower courts, the Cours des Bourgeoise and the feudal courts.