A first guardian is an entity that exists to protect its planet and "facilitate the planet's ultimate purpose".
Known First Guardians
History
First guardians are typically almost as old as the planets they guard, and they are born through weird plot shit and time shenanigans. Every planet destined for intelligent life has such a guardian, and though their forms vary widely, they all share a common and potent genetic code. This code grants them near omnipotence, and they can be "merged with a host of great intelligence", granting them near omniscience as well. Doc Scratch is Alternia's first guardian, Becquerel is pre-scratch Earth's first guardian, and the God Cat is post-scratch Earth's first guardian. Beforus had a different first guardian as well, but it has yet to make an appearance in the story. If current trends are indicative, first guardians have a pure white body and a unique "green energy" effect, in which they appear to flicker and flash with a green electricity, appearing to shift in and out of existence and generally radiate energy. Another object with similar properties is the Green Sun, the source of the first guardians' powers.
It seems their ultimate purpose is to help complete stable time loops, in order to have the game developed and played. Becquerel demonstrates this when he leads
Jade to John's gift, allowing them to meet in the first place. Doc Scratch manipulates the trolls through Vriska, and at one point uses her to orchestrate the death of Aradia. He finds Aradia much more convenient to manipulate, controlling her into tricking Sollux into developing the game, and presumably also controlling a number of events within their session.
A secondary purpose of first guardians is to ensure their own existence through stable time loops. Bec influenced his own creation by protecting Jade and taking her to John's note, though most of his creation was facilitated by others such as Jack Noir and the trolls. Doc Scratch brought about his own existence as well. God Cat saved Jane from the letter bomb left for her by the Batterwitch by teleporting her away, and she was later seen apparently preparing to create GCat while under the Condesce's control. GCat also caused Jaspers' death by momentarily appearing in Roxy's room, which could imply that his ectobiological parent may be Jaspers, and in getting Jaspers killed, GCat was ensuring that he is paradoxifiable – a requirement of ectobiology. It is unclear if there's a consistent pattern for first guardian ectobiology: GCat and Becquerel were both created with a combination of the genetic code and an animal, whereas Doc Scratch was created with two inanimate objects, in addition to the genetic code (which, in his case, was strung together through five separate sources). It's possible that the presence of Caliborn's soul in Lil Cal is what made him a valid ectobiological parent for Doc Scratch.
In addition to his duties regarding Sgrub, Doc Scratch is working to ensure Lord English's entry into the Alternian universe. Although it's possible that every first guardian works for Lord English, this seems unlikely, and Doc Scratch's situation is more probably unique, possibly brought about by Lil Cal's connection with Caliborn. Doc Scratch contacts Rose and supplies her with information, seemingly in order to destroy the Green Sun and kill himself, thus allowing Lord English entry into that universe.
Dog-tier Jade in many ways acts as Earth's First Guardian (in part because she now has first guardian powers, and the previous first guardian only exists as part of her) - she is responsible for the relocation of Earth required for the Cherub session.
Another theory is that they are related to Calliope in some manner, due to their green-tinted energy, their guardianship (the most passive class would literally help everyone but herself, similar to how the most active would help only himself), and their rampant use of spacial powers, such as teleportation of themselves and other objects, as well as their proclivity towards other space players (Jade and Kanaya). Another similarity is Lord English's arrival by usurping the body of Doc Scratch, similar to Caliborn's usurping of their shared body from Calliope.