Multipliers in Age of Mythology and Age of Empires III are increased bonus damage versus another unit. Instead of an additive increase (i.e. +32 Damage versus Cavalry) that was present in the Genie Engine era games, 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 now shows the multipliers units have against one another.
The functional effect of this is "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 are called "Hard Counters," meaning that by default they are likely to survive 1-on-1 without taking much damage. "Soft Counters" do not have bonus damage versus units (or at least an amount smaller than x1.75).
If an Archer (or Light Infantry in Age of Empires III) has a 5x multiplier against Infantry, then it takes the unit's base damage and multiplies it by 5 to heavy infantry. Units may also have negative multipliers as well. If a unit has a 0.5 multiplier against Villagers then it does only half of its base damage to villagers.