Illusions are a cosmetic enchantment sometimes found on Weapons. Illusions can be removed and applied to other Weapons.
A Weapon with an Illusion will display a unique weapon skin. Each weapon type has several Illusions. Examine the appearance of your weapon by pressing Z.
Illusions are tied to weapon type. For example, Bardin's Great Axe can have several specific Illusions that only apply to that weapon. Each character has two basic illusion types (For Bardin it's "Beardling's" and "Warrior's") for each weapon, and then several unique named illusions. There are several illusions that only appear on red weapons, these have unique names and are usually more impressive and usually have a teal glow in the details.
Some illusions currently appear to be the same as the default skin, and some share the names of default weapons. It is unknown if this is a bug.
For a list of all known illusions check out the Weapon Illusion pages