The Ottoman Empire was divided according to a series of agreements and treaties both between the major powers as well as with the representatives of various groups and interests.

Some of these are well known like Sykes-Picot and the Balfour Declaration, others are less well known like the Treaties of Sevres and Lausanne.

It's worth pointing out that your question is really two, 1. how the borders were determined, and 2. what the identity of the states within those borders would be.

The process by which this was accomplished is pretty enormously complex, so it might be worth pointing out what the major accomplishments were of each initiative, chronologically.

-- The DeBunsen Committee, 1915: The DeBunsen Committee was convened in 1915 to determine what British war aims against the Ottoman Empire should be in the event of war. While the conclusions of the committee were not part of any binding treaty, their findings, that the Ottoman empire should be partitioned with particular zones of control, held remarkably true to the final distribution of formerly ottoman territory.

-- The Hussein-McMahon Correspondence, 1915-1916: Forming less of a treaty than an understanding, the result of these negotiations between the British government and Sharif Hussein of Mecca's son Abdullah sought to gain the support of the Arab peoples under the leadership of Sharif Hussein and the Hashemites in exchange for a promise of territorial control of most of the Ottoman Arab Middle East. There were some territorial reservations, particularly regarding French areas of influence in Western Greater Syria, but the difference between the apparent scale of the territory promised to the Arabs to be ruled by the Hashemite dynasty and what they actually received is the result of endless controversy and conflict ever since, largely as a result of:

1. The Sykes-Picot Agreement, 1915-1916: This war time measure negotiated between the British, French and Russians sought to divide the Middle East into zones of control and influence. Despite its infamy, some of the most problematic divisions of the Ottoman Empire (e.g. the composition of Iraq, the status of Palestine) described by Sykes-Picot did not come into being. It's reputation in the popular imagination is less the result of what it actually accomplished (i.e. Sykes-Picot did not actually confirm the territorial division of anything, it was a provisional wartime understanding) than that it was evidence of imperial double dealing with the Arabs.

and 2. The Balfour Delcation, 1917. This document, the only one of the above agreements to be published and made public (Sykes-Picot was revealed by the Bolsheviks following the Russian Revolution) promised a "national home for the Jewish people" in Palestine, which was to be ruled as a British mandate.

Following the conclusion of the war, the British and the other parties involved had to find a way to rectify these competing and contradictory understandings about how the former Ottoman territories ought to be divided.

In terms of territory the results were largely determined by the Treaty of Sevres (1920) and the Treaty of Lausanne (1923). The primary difference between the two was that where Sevres would have gutted the Turkish heartlands in Anatolia between the Allied powers, Lausanne maintained the territorial integrity of Turkey's modern borders (with the exception of Hatay province, a small, then predominantly Arab part of Syria). Lausanne also resolved the so-called Mosul question about whether Mosul province should be part of Iraq, Syria or Turkey. For a number of reasons, which I can touch on below, the British wanted it to be part of Iraq and had the strongest bargaining position. Also as a result of military occupation it was defacto controlled by the British throughout the time period anyways.

That more or less covers the territorial borders of the former Ottoman Empire, but the character is a different question.

After the war the British had a hell of a time reconciling their promises to the Arabs with their promises to the French and the Zionists, not to mention the American demands that post-war settlements should be made in accordance with the wishes of local peoples in line with their right to self-determination.

Which is why at the Paris 1919 peace conference the major powers essentially punted. The French (falsely) denied any knowledge of the Hussein-McMahon agreement but all agreed to put the final decision of what to do with Greater Syria (including Palestine) up to the result of a study. The result was the King-Crane Commission, which produced fascinating material about what the Arabs of Syria wanted in terms of their own government but was totally ignored following Wilson's incapacitation from a stroke in 1919. Their findings weren't published until 1922.

Rather, the identity of at least the British portion of these states was determined in large part by the Cairo Conference of 1921. There it was determined that the Hashemites would rule Hejaz, Iraq and Transjordan.

French policy in Syria was a bit more hamfisted. As a matter of longstanding policy and discussion with the Maronites the French wanted to create a Maronite state in the area of Lebanon. This they expanded to the largest geographical area possible that would still be majority Christian and the result is the modern borders of Lebanon. The problems there largely result from the fact that Christians (in all likelihood, a real census hasn't been conducted in nearly a century) have not been the majority there for some decades.

In the rest of French mandatory Syria, the French initially tried to divide the country in four along ethno-religious lines with an Allawite state, Druze state and separate state for the areas around Damascus and Allepo. Following a large rebellion against the plan the French ruled Syria as a unified entity across ethno-religious lines. That the French wanted to create states reflecting ethno-religious divisions is painfully ironic given our contemporary criticism of the imperial division of the Middle East.

Policy in Palestine was determined in large part by a series of whitepapers largely concerning the extent and attitude towards Jewish immigration into Palestine. These are the Churchill White Paper of 1922, the Passfield White Paper of 1930 and the White Paper of 1939.

Questions relating to the partitioning of Palestine came most prominently to the fore with the Peel Commission and the UN partition plan but that's probably a separate topic.

I think I mentioned Iraq, which is of course a sticky subject, and the Mosul question. There are a few reasons why Iraq looks the way it does. 1. Ethno-religious balancing: the British thought they could balance Shia against Sunni, Arab against Kurd, with Faisal and the Hashemites unifying the country but being difficult enough to require an indefinite British military presence. A decent idea, but the collapse of the British economy led to the end of the Iraqi mandate in 1931 and the end of Hashemite rule would eventually lead to Saddam Hussein and Sunni-dominated Baathists. It's worth remembering here that Iraq was a wildly different country, demographically, in the 1920s than it is today. 20% of Baghdad was Jewish, it had huge Chrisitian populations, etc. etc. So this idea that they should have just created Sunnistan, Shiastan and Kurdistan that sometimes pops up today is quite anachronistic. 2. Oil, Mosul had oil, the British wanted it. 3. The original idea in Sykes-Picot was that Mosul would be French because the area to the North would be Russian. Britain didn't want to share a border with Russia. 4. Defacto control: the British were there first and stayed there. 5. Faisal and his clique, after 1921, wanted Iraq to be as large as possible for their own benefit and ambitions. That meant they advocated for controlling Mosul as well.

Phew, lot of text. I hope that's helpful.

Source wise: wow, lots. There's some great primary sources related to all of these texts that are worth looking at that should be quite easy to find (e.g. the texts of Hussein-McMahon, the maps associated with Sykes-Picot.) For a longer explanation of this I quite like Western Imperialism in the Middle East, 1914-1918 by D.K. Fieldhouse. Eugene Rogan's The Arabs: A History also covers much of this quite well. George Antonious' The Arab Awakening is probably the most iconic depiction of the betrayal of the Arabs by the Western powers. I hesitate to call it "biased" it's more like a primary source (Antonious was a direct party to the events and published the first English translation of the Hussein McMahon correspondence.)

I always struggle to find a link but there are also some fantastic and fascinating transcripts of the 1919 Peace conference on these issues.