Morale was a mechanic in the Animus which tracked guards' motivation to fight in open combat. The values only applied once the user entered open combat, so any action performed before the initiation of combat did not affect the morale of the guards.
The Animus 1.28 used by Abstergo Industries, had a basic system of morale. When Desmond relived Altaïr's memories, the lowest ranking guards would easily be scared by the deaths of their comrades, or superiors.
Guard Sergeants were much harder to scare, and would only probably run away if the leader, was killed.
Guard Captains were the bravest of the regular guards. They would hardly ever be scared away, no matter what Desmond did. However, if one of these was killed however, all the other lower ranking guards would be terrified, and probably run away.
Templar guards were a unique guard archetype, the strongest and bravest of all the guards. They would never run away, regardless of the deaths of nearby guards. As these men were held in very high regard, whenever one of these duelled an opponent, the other guards would stand back, and watch. The death of one would completely shatter the morale nearby guards, few of whom would stand and fight.
When Desmond relived Ezio's memories, the Animus used by the Assassins had an updated Morale system. The Animus 2.0 specified morale points for each guard archetype.
Brutes had 1000 morale points in the Animus, and they never ran away unless disarmed and completely unable to find another weapon or get an improvised one. Militia guards had 20 morale points, and were the most cowardly guards, likely to run away from a battle if it ever got too intense.
The death of any guard caused a drop in morale of all the other guards. The death of any lower ranking guard caused a drop of morale by 5 points for all surrounding guards. A Counter-kill, Disarm, or ground-assassination also caused a drop of 5 Points. If a disarmed guard himself couldn't find a replacement for his weapon, he would immediately run away. Disarming an opponent and killing him with his own weapon reduced morale by 10 points. Killing a Seeker, or killing an enemy using a pistol was worth 20 points, and killing a Brute worth 30 points.
Every time Ezio took a hit the guards' Morale rose by 10 points, and if Ezio reached critical state, the guards would gain 25 points each. If Ezio ran away from a confrontation, it would add 25 points to every guard's Morale. At the Auditore Villa, Desmond could get Ezio to learn the skill of demoralising an enemy by pointing a pistol at him.
When Desmond began to relive Ezio's memories in Rome, the morale system was slightly updated. The guard archetype that appeared in the other cities Ezio visited had the same Morale stats that they had in previous cities.
Two new enemy types are introduced in Rome the Papal Guard and the Borgia Captains. The Papal Guards never ran away from a fight. Conversely the death of a Papal Guard would frighten away almost every guard that was fighting alongside him.
There were two types of Borgia Captains, ones who ran away from Ezio upon seeing him, and ones who stood and fought. The death of a Borgia Captain would terrify every single guard that witnessed it, regardless of whether or not they were engaged in combat alongside their leader.
Additionally, a new feature was introduced in the Animus. Certain weapons that Ezio used during his travels could scare away enemies. In Rome the only such weapon was the Dagger of Brutus. This dagger had different, and more brutal kill animations compared to regular daggers, causing a drop by 20 points in most guards morale. Most lower ranking guards who witnessed it ran away.
When Desmond fell into a coma, he was put into the safe mode of the Animus, which kept his body alive. The software had been updated to the 2.03 version by then. While he explored Ezio's memories from Constantinople, the morale system was largely the same as earlier. Papal Guards were replaced with Janissaries, and Borgia Captains with Templar den captains, whose morales were largely the same as their Roman equivalents.
Two new pieces of equipment were introduced, that caused a morale lowering effect similar to the Dagger of Brutus. The Sword of Vlad the Impaler, as with the Dagger of Brutus, had brutal kill animations, which scared guards away. The Armor of Ishak Pasha, was an unbreakable piece of armor, and was said to bear a cursed marking. Hence any soldiers that saw it would get scared.
An additional feature related to morale was introduced in the Animus 2.0. Whenever any guards ran away from a battle, it would increase the Notoriety of the user.