The Gnosis is a highly eminent and divine object, a tool that the Archon's possess as a way to connect & resonate with Celestia, the abode of the Archons. The gnoses are all shaped as chess pieces, with Barbatos' gnosis being shaped as a Queen, & Morax's being shaped as a Rook. Currently, these are the only gnoses with explicit showcases of their appearance. A gnosis is a proof of an archon's position as an elemental god of Celestia, being given to them by Celestia.
A gnosis is not a source of an archon's strength or abilities. Barbatos, Morax, & Raiden Ei have all displayed the ability to wield their element's power without the possession of a gnosis. Of course, a gnosis would greatly enhance the strength of their powers. Interestingly, loss of an archon's gnosis would also mean they gave up their status as an archon; being their resonance and connection to Celestia would have been severed.
As it stands currently, 4 of the Seven have given up their gnoses to the Tsaritsa, the ruler of Snezhnaya. The Fatui, (or Fatuus) & the Tsaritsa seem to be looking to confiscate the gnoses' of every Archon, for a rebellion against the divine that the Tsaritsa is plotting. Barbatos, Morax, the Tsaritsa herself, & Beelzebul (the latter having long cut ties with Celestia & giving her gnosis to their companion) have all given up their Gnosis; whether it have been by force, by choice, by conviction, or by burden.
Venti’s Anemo Gnosis, the queen piece, in her possession. Stolen by La Signora near the end of Genshin Impact’s prologue chapters.
Zhongli’s Geo Gnosis, the rook piece, which was handed to La Signora on the basis of a contract signed with the Tsaritsa. This special contract is the “contract to end all contracts”, what exactly the Tsaritsa offered is unknown.
The Tsaritsa's own gnosis is a given.
Beelzebul, or Raiden Ei's gnosis is currently in the possession of Scaramouche, the Balladeer. Also originally Sixth of the Fatui Harbingers. Originally, this gnosis was to be contained by a puppet engineered by Ei; however none of these puppets could bear the power & responsibility. As a result, the Head Shrine Maiden of the Grand Narukami Shrine, Yae Miko or Guuji Yae was given the responsibility of the gnosis.
The remaining 3 Gnoses are expected to be taken as well, if the Tsaritsa's plan remains persistent. The fate of these trinkets are yet to be known.
Fatui Emblem. CR: Genshin Impact miHoYo Co., Ltd.
The Gnosis is seemingly possessive of very peculiar properties. It's described within the game that, as long as the gnosis is in the Archon's possession, the Archon themself has an inalienable, divine ability to protect their region.
Interestingly, also most likely shares the indestructibility of regular Vision, being they're divine "treasures" & (most likely) possess the same elemental properties.
The Gnosis also appears to act as a communication device for archons to Celestia, being Venti describes them as magical focus' that resonate directly with Celestia.