Multipliers are factors used to determine bonus damage against other units in Age of Mythology and Age of Empires III. Instead of an additive increase (i.e. +12 damage against cavalry) that was present in Genie Engine games, the Bang! Engine increases the damage by multiplication.
Bonus damage is not explicitly referred to in Age of Mythology and can only be found by using the prototype definitions, a unit's rollover description. In Age of Empires III, the information panel shows the multipliers units have against one another.
The functional effect of this is a 'rock, paper, scissors'-type gameplay where each unit has certain weaknesses and strengths that balance other units to avoid major edges during a battle.
Units that have multipliers against certain units are often referred to as counters to those units, meaning that by default they are likely to survive 1-on-1 without taking much damage.
If an archer unit (or light infantry in Age of Empires III) has an attack value of 6 and a 5x multiplier against infantry, then it deals 30 base damage to infantry. Units may also have multipliers reducing the damage output (values between 0 and 1). If said unit has a 0.5 multiplier against Villagers, then it only deals 3 base damage to Villagers. Note that the actual amount of damage dealt may differ due to the target's armor.