Equipment

Armour:

Unlike the Western European Crusaders, Muslim Soldiers never wore fully body armour. It is without a doubt a fact that Europeans were more heavily armoured than Muslims but that was mainly due to different fighting styles used by both sides. Muslims relied heavily on mobility of their famous horse mounted archers and hence preferred light armour to increase speed and agility.

According to D.C. Nicolle's Arms and Armour of crusading era 1050-1350, Muslim soldiers used following Armour pieces:

1. Head: Turbans are often wound around presumably conical helmets. Other headgear worn included fluted helmets, low or tall conical helmets without turbans, mail coifs, and brimmed hats or war hats (pages 327 and 806). Conical helmets had the distinct advantage of deflecting straight blows to slip, reducing the damage.

2. Torso: Muslims also used mail hauberks as their primary defense in hand to hand combat, and for their heavy cavalry. It seems, however, that the bulk of the Muslim army consisted of light cavalry which may have worn only a padded overcoat instead of the mail hauberk (XIX Chapter).

3. Arms & Legs: Large riding boots or overshoes were worn by the Islamic cavalry. These were loose enough to fit a small weapon inside. They were often worn in conjunction with the Saq, the Islamic version of the chausses. Cavalry wore additional leggings above the boot known as Ran, a piece of armour comparable to the European cuisses. These were tied to the waist-belt and could be made of iron as well as other materials (Pages 618-619).

4. Shield: Shields were predominantly round in shape with a boss in the middle. The choice of smaller and lighter shields was again to prefer speed over protection.

5. Horse Armour: The horses of the Islamic armies wore a quilted covering called a Tijfaf or Bargustuwan for protection (Pages 587 and 624).

Weapons:

Bow: The Muslim forces were experts at using the curved bow and the composite bow. Much like the English longbow, it could only be handled effectively by someone who had trained and developed the necessary muscles to wield it. Unlike its English counterpart, it was used as an offensive cavalry weapon and it had a more penetrating and longer target range. (The Oxford illustrated History of Crusades by J. Riley-Smith page 232)

Sword: During the time of the Crusades, the Muslim forces used both straight edged and curved swords. The Mamluks, at one point known as best cavalry in the world, exclusively used curved scimitars. Other than that usual knives and daggers were also used. [Reference]

Spears: Muslim Forces used variety of Javelins, Spears and other weapons of this family.

Incendiary Weapons:

Facing the power of fire in warfare at hands of the Byzantines, Muslims developed their own incendiary weapons too which were effective both on land and sea. They usually used naptha, a type of clear gasoline. Naptha was then used in weaponized form in primitive flamethrowers or firebombs fashion. [Ref: Islamic Weapons, warfare and Armies by Paul Hilliam].

Siege Equipment and Artillery:

1. Ladders: The most basic tools were ladders but they allowed the defenders to push them over. The best time to employ this tactic was when the attackers had the advantage of surprise or when there were an overwhelming number of them.

2. Counter-Weight Trebuchets: These were used to bombard enemy fortifications to rubble. First mention of this device is found al-Tarsusi's warfare manual, written for Ayubid Army.

3. Mangonel: This was used by both Muslims and Christians. The Crusades: Islamic Perspective by Carol Hillenbrand. (Page number 533).

4. Siege Towers: Siege towers were used extensively to penetrate the walls of fortresses by both sides.

Tactics

Although many standard practices were shared by both sides, Here we will discuss only the ones which were characteristically Muslim.

Quoting from The Crusaders in Syria and the Holy Land by R.C. Smail (page 64-87):

The main determinant of the Muslim tactics during the crusades was the predominant position of the horse archer in their army. As a light cavalry, the horse archers that made up the bulk of the Islamic armies were highly mobile. This mobility was used in four ways which gave the Turks an advantage over the European armies which could be highly dangerous when they approached for close combat. High mobility allowed the Islamic army to maintain a distance from the enemy and choose the moment at which they would close with them. This allowed for the Muslim forces to generally control the place and timing of the major confrontations. The second use of this mobility was the feigned retreat which allowed them to lead the Europeans into ambushes or to cause a fake retreat to last for some days to tire the enemy. Good mobility allowed them to attack the weakest points in the enemy's army. The rear and the flanks made ideal targets against the Europeans since the commander would travel in the vanguard at the front of the column while traveling. This caused chaos while traveling, and in a battle it allowed the Turks to attack the crusaders without ever coming into a pitched battle with the main body of the Europeans until they were worn-down and had lost all support from the flanks of the attack in battle. The last way in which this mobility helped the Muslims was in attacking the enemy and forcing him to fight on the march. Instead of the Europeans sitting in a circle and gradually being killed by arrows, they marched toward either safety or where they supposed the enemy to be. This allowed for even greater possibilities of the use of the Turkish mobility in battle to come into distance, attack with arrows and move back away from the distance in which the slower loading crossbow could be used against them.

The horse archers themselves could loose their arrows from the saddle without halting or dismounting, and even shoot backwards while in retreat. The composite bow was a light weapon which often did not penetrate the armour far enough to cause injury to the wearer. Due to thick pads of felt and the mail worn by knights, the arrows were often seen sticking out of warriors who just continued on their way. The many flights of arrows were used mainly to destroy any cohesion that the Europeans would have, and cause them to lose horses as well. This loss of horses was a major blow to the main weapon of the Crusaders; namely the heavy cavalry charge with lances in rest. The horse archers would begin the fight at bowshot range patiently waiting for the most opportune time to attack at close quarters and finish the enemy. Only when a favorable opportunity arrived, or when they were forced into confrontation would the lightly armoured Turks attempt combat at close quarters.

This is a very broad and interesting question. I have tried to answer it as best as I could. I hope you find it informative.