Damage is one of the 5 attribute stats that help to determine success or failure in combat against creatures and other players.
The other attributes are Attack, Armor, Defense, and HP. Some of the older items in the game also have a sixth attribute, Stamina.
Damage is related to the amount of Hit Points (HP) that are removed from an opponent during a successful hit. The more damage a player has, the more HP are removed with each hit. Damage can be countered with armor.
The exact formula is unknown, but this is an approximation:
A player has 700 points on Damage
A creature has 550 HP and 300 Armor
Approximately 300 of the player's damage will be absorbed by the creature's armor and the remaining 400 damage points will remove 400 hit points from the creature. This would leave the creature with 150 HP and the next hit by the player should defeat the creature.
Again, this is only an approximation of the formula. Each attack can take more or less HP from a creature with the same stats according to a formula that is likely based on percentages.