As mentioned in Outremer: A Brief Time of History, the regaining of Edessa with the Second Crusade was the key to Outremer's central idea of the Crusader States lasting into the Renaissance. In real life, Edessa fell to Nur ad Din from Aleppo and Damascus, and was thus the first of the Crusader States to be regained by the Muslims. In the Outremer Scenario, the Second Crusade sent to regain Edessa from Nur ad Din was eventually a success, though King Louis of France and Count Jocelin II of Edessa died, the crusaders rallied around Louis' young widow, Eleanor of Aquitane, and drove off the Syrians, Nur ad Din dying in the battle.
Eleanor proclaimed Edessa to be a sovereign Principality, rather than a County of Jerusalem, owing fealty to no-one. As Princess of Edessa, she wed Henry II of England, and the Principality eventually was given to her son Richard Courleon, and through him to his second son, Bohemund. The heirs of Bohemund still hold the throne of Edessa in 1560.
The court at Edessa is Angevin rather than English, and a peculiar idiosyncratic French - called Edessan - is the language of choice. It was sundered from French early on, and has a lot of antique word formations, as well as a lot of Arabic and Turkic loan words. Turkish is spoken mainly in the country, as well as Kurdish. Armenian is also spoken widely, as a good number of Armenians have settled in Edessa.
Richard began a process of inviting in Islamic splinter groups who were being persecuted in their own lands. The Alevi, a Turkish and Kurdish offshoot of Shi'a, in particular, settled heavily in Edessa under the protection of the Angevin overlords, and eventually gained great wealth and status. There are also a great number of Druze from the Lebanon and Anti-Lebanon mountains as well, with smaller numbers of other sects.
The Principality is, in essence, a strong alliance of Angevin-descended native urbanites, rural Alevi and Druze chieftans, and Armenian merchants. There has been a great amount of intermarriage despite the religious differences, and the lines between these groups are rather blurred and mostly theoretical by 1560.
The Principality is extensive, fertile, and well watered, unlike most of the Outremer States. The great Frankish horses are raised here and sold throughout the Outremer. They have been interbred with Arabian horses, with various interesting breeds being raised. Food is exported as well from the rich farms of Edessa. Cattle and sheep are raised in pasture lands, and wool is a prime export. Some small scale manufacturing is done in the cities, but is not a prime concern.
Edessa has traditionally had poor relations with Antioch, and good relations with Armenia. It's relations with Aleppo to the south and the Turks to the north vary more widely, good at times, warlike at other times. Edessa has even allied with Aleppo against Antioch a couple of times down the centuries. The natural trade routes go from Baghdad in the east to Armenia in the west, bypassing Antioch entirely.
Here's a map!