A first guardian is an entity that exists to protect its planet and "facilitate the planet's ultimate purpose".
Known First Guardians Edit
First guardians are typically almost as old as the planets they are born on, and they are born through Weird Plot Shit and Weird Time Shit. 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. Pre-scratch Alternia 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 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 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.
In addition to (or possibly as part of) his duties regarding Sgrub, Doc Scratch is working to ensure Lrd English's entry into the Alternian universe. It's possible that every first guardian works for Lrd English, though it's more likely Doc Scratch's situation is unique. He contacts Rose and supplies her with information, seemingly in order to destroy the Green Sun and kill himself.
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 Lrd English's (re?)arrival by usurping the body of Doc Scratch, similar to Caliborn's usurping of their shared body from Calliope.
After the Scratch, Earth's first guardian appears to be the mysterious God Cat, a feline living in Jane Crocker's neighborhood. The cat saved Jane from the letter bomb left for her by the Batterwitch by teleporting her away. God Cat's genetic origin may potentially have something to do with Jaspers and/or Mutie.