During quests, an Adventurer may equip a skin over their weapon, giving it the appearance of another weapon. The stats, skills, and abilities of the "real" weapon remain unchanged - for all intents and purposes, this is purely a cosmetic feature.
On the teams screen and in co-op rooms, units with an equipped weapon skin will have a blue eye icon displayed next to them. Tapping a unit in either place will then show the name of the "real" weapon in the unit's summary page.
Weapon skins can be disabled from view in co-op rooms by going to Game → Weapon Skin Display Settings in the More (Green Book) → Options menu.
For playable weapons that actually have stats and such, their skin is unlocked upon obtaining that weapon.
For weapons that only exist as skins, they are usually unlocked when unbinding a playable weapon. The skins available will vary by weapon, but will usually match the upgraded weapon's type and element. For example, unbinding a Flame Sword may unlock some other Flame Sword skins.
Special skins may be available as rewards from events.
A running count of the total available skins can be found below. Note that these counters include both playable weapons and skin-only weapons:
To equip a weapon skin, go to the teams menu, tap an adventurer's equipped weapon, and then tap the "Weapon Skins" button in the bottom-right to select a skin. Note that skins can only mask weapons of the same weapon type: for example, a lance can be masked by another lance, but a lance cannot be masked by a wand.
Swapping the "real" weapon to a different one will instantly unequip whichever skin was currently in use.
During the Walkies quest of the Wagabond Pupper event, all adventurers who are eligible for weapon skins will instead carry a small shovel. Adventurers that cannot use weapon skins will still carry their usual built-in skin. For example, Veronica will still hold her tome.