Will represents the courage and determination of a character. For example, it might determine if a character continues his quest or quit after facing difficulties. It also determines if a character faces his fear or runs away. It is used to resist special effects such as charms or paralysis.
A will check is required when the character faces an especially big shock, such as a crushing blow (losing more than 50% of his total HP) or a mortal blow (bringing his total HP to zero). Also, a check might be required if a catastrophe kills all the character's companions, leaving him alone. In this case, a successful check means the character doesn't succumb to madness. Finally, when a character falls into one of his threshold (HP or MP), a will check is required or the character falls on the ground, drained and exhausted.
Will also provides a bonus on critical spells, allowing better chance to control the magic to obtain greater effects.