Morale properly refers to the state of mind of "troops" during combat or stress situations. Stupid creatures tend to fight to the death. So do creatures with a set purpose in mind — elite, guards, and fanatical creatures. (PHB.106)
Your character will never have to check morale status, nor will any other player character, for each player provides this personally. Some are brave, some foolish, some cautious, some cowardly. Your character's henchmen will probably have to check morale, so too will hirelings.
Powerful monsters will never check morale, and even weak ones will probably not do so as long as they have leadership. (PHB.106)
Morale checks are used to determine the amount of will to fight in non-leader NPCs, and can be applied both to henchmen and hirelings of character types and groups of intelligent opponent monsters.
The base unmodified morale score is 50%. [Creatures that roll this number or LESS succeed the morale check.]
The creature is faced by an obviously superior force (e.g. opponents are hitting twice as often): Check each round
25% of the creature's allies are eliminated or slain OR the creature suffers 1/4 h.p. damage: Check at +5%
The creature's leader is unconscious: Check at +10%
50%+ of the creature's allies are eliminated or slain OR the creature suffers 1/2 h.p. damage: Check at +15%
The creature's leader is slain or deserts: Check at +30%
Other modifiers to a Morale Check roll:
Bonuses
For each enemy that deserts/flees: -5%
For each enemy slain: -10%
For inflicting enemy casualties without suffering any: -20%
Penalties
For each allied creature slain: +10%
For allied group suffering casualties without any enemy casualties: +10%
For each ally deserting: +15%
The creature(s) is outnumbered by 3 or more to 1 (or significantly "outclassed", per DM): +20%
Henchmen or Associated Creatures. For each creature take the base morale (loyalty) score and cast percentile dice. Adjust the score for penalties and bonuses, and if the adjusted score is equal to or less than the loyalty score, morale is good. If the number exceeds the loyalty score, the creature will react according to the morale failure table below. (See Non-Player Characters, Loyalty, Obedience, and Morale.)
Monsters, Intelligent. Each monster has a base morale of 50%, +5% per hit die above 1 HD, +1% per hit point above any hit dice. e.g. A troll with 6+6 HD would have a normal morale score of 81%.
If a morale check is required, scoring and reaction are as stated above for henchmen et al.
The check is missed by 1% to 15%: The creature will fall back, still fighting.
The check is missed by 16% to 30%: The creature will disengage or retreat.
The check is missed by 31% to 50%: The creature will flee in panic.
The check is missed by 51% to more: The creature will surrender.
The difference between the maximum score needed to retain morale and the number actually rolled, as adjusted, is the % of morale failure.
[See COMBAT, Avoid or Flee, Pursuit and Evasion.]