This Relic reduces the chances of getting a Common Drop.
As a result you have a higher chance of getting any other rarity (also Legendaries).
This Effect does stack if more people use this Relic.
Information about drop chances in Borderlands 2 (Forum Post before Legendary drop rate change in May of 2015)
Like most Legendary Artifacts the Vault Hunter's Relic (BL2) has one random Variable Ability out of these 4 Categories.
The Vault Hunter's Relic (BL2) is not a Standard Legendary Artifact.
It doesn't have a Variable Ability like most Legendary Artifacts.