Multipliers in Age of Mythology and Age of Empires III is an increased bonus damage versus another unit, instead of an increase (i.e +32 Damage versus Cavalry) that was present in Genie Engine era games, Bang! Engine instead make use of increasing the damage, by multiplication, rather than extra damage.
Bonus damage is not referred to explicitly in Age of Mythology and can only be referred to by using the prototype definitions, a unit's rollover description. In Age of Empires III the information panel now shows a multiplier, one unit has, against another.
This function works the same way as rock, paper and scissors where each unit has certain weaknesses and strengths that balances 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 that much damage, whilst units that are called "Soft Counters" do not have bonus damage versus units (or at least an amount smaller than x1.75)
If a Archer (or Light Infantry in Age of Empires III has a 5x multiplier against Infantry, then it takes the units base damage and multiplies it by 5 to heavy infantry. Units can also have negative multipliers as well. If a unit has a 0.5 multiplier against Villagers then it only does half of that base damage to villagers.