Units with Awe force enemies to take a morale check against 10 + Awe in order to be able to attack them. Thus, a unit with Awe +4 would force attackers to pass a morale check against 14, or be awestruck and unable to attack the awe-inspiring monster on that combat round, meaning that the unit flashes white and nothing happens.
Awe may be bypassed via ranged attacks (arrows, magic), mindlessness, berserk, or high morale. Awe does affect Trample, and trampling units may have to pass separate awe tests for each unit in the square to be trampled.
Awe may be obtained from multiple sources, and stacks additivly. Thus, a Virtue (Awe 5) with Dominion 10 (Awe 3) will have Awe 8.
Items such as the Shield of Gleaming Gold
Certain Heroic Abilities
The Throne of Splendor
Some units, such as Siddhas, start naturally with Awe
Xilbaban Sun Guides while Theft of the Sun is active
Moreover, there exists a parallel ability called "Animal Awe" that has the same mechanic, but only affects animals. It does not combine with awe, but a unit may have both active at once, and an attacking animal will have to pass both effects.
*Note that the manual provides incorrect values of 1 and 2, as well as an incorrect calculation.