Who was the main Muslim army leader during the Third Crusade?
The main commander of Muslim forces during the Third Crusade was Saladin. Saladin had been trying to reduce the Crusader states since the end of the Second Crusade. While he was chipping away at the remaining Christian strongholds, a new Crusade was called.
During this crusade, Richard the Lionheart was able to capture many coastal cities despite the best efforts of Saladin. The two fought to a draw several times with each coming to respect the other. Eventually Richard signed a treaty with Saladin that allowed Christian pilgrims to travel unimpeded to the Holy Land.
Saladin had failed to drive the invaders out, which disturbed many in the Muslim world, but he also kept control of Jerusalem despite another huge invasion.
By the way for a more neutral, balanced contemporary account of Crusaders and Crusades (seen through 'Muslim eyes') it might be good to read something of the memoirs of the prince Usama ibn Munqidh of Shaizar, a useful primary source and an interesting historical document.
The Sultan of Egypt and Syria, founder of the Ayyubi/Ayybid dynasty, the strongest and most influential Muslim leader during the Third Crusade, was the Sultan Salah-ud-din Yusuf Ibn Ayyub (born c. 1137/38-died 1193 AD), better known as 'Saladin'; and we might consider him the 'main commander' of the forces of Islam at the time. It was under his personal leadership that the Muslims were able to recapture the parts of Palestine that had been captured by the Christians/Crusaders of the First Crusade, some 88 years before.
It must please be kept in mind that for Islam/Muslims, at that time, there were a large number of sultanates and kingdoms etc, in North Africa, the Middle and Near East, all the way to India and Central Asia; and at no time was the struggle of the 'Crusades' (taken so seriously in Western Europe) a major or very important struggle -- the Crusaders or 'Nasrany' (i.e. Christians/Nazarenes) were at best considered a nuisance who had captured and annexed some areas. The only important concern was, the capture of Jerusalem, where apart from Jewish and Christian holy monuments, there was also the Dome of the Rock and al-Aqsa Mosque, the Third Holiest site in Islam. The various Muslim rulers and people of the region, thus, were only worried that the Crusaders (whom they considered savages and brutes) would not only damage or desecrate this shrine, but also stop Muslims from making pilgrimage there. This was the main concern prompting the Sultan Salah-ud-din too, and he had vowed to free the area from the bigotted and often rapacious Crusaders, and ''open Jerusalem once again to all creeds''(Anthony Nutting, The Arabs, 1964, pp.181-182).
In fact, as the Crusaders were already being rolled back in the whole region , only a small islet of 'Latin Palestine' held out, and despite all, Saladin assembled a huge force and all the vast resources of Egypt as well as of some of his allies, to crush this kingdom decisively once and for all at the battle of Hattin, July 4th 1187. Jerusalem itself was taken in September of that year.
The subsequent 'embarkation' of King Richard of England and Philip of France et al, on a Third Crusade to supposedly 'rescue' Jerusalem etc, was a futile and expensive adventure, yielding little of substance. In 1192, before going back home King Richard and Saladin amicably signed a treaty to demilitarise a large chunk of Palestine and to maintain Jerusalem as an open city for all.
In the second link below, you can see the main/actual 'field commanders' employed under the Sultan Saladin's overall command.