"The One who would be Guardian ascended the throne -- witness the mural -- where her dreams and hopes, her very soul were locked away in the disc. In service of the Balance, these traits were but barriers. Through new eyes, the imbalance between the worlds was as clear as the stars themselves to the Guardian and with one thought ... she channeled chaos from Arcadia and logic from Stark into the disc and redistributed the power wherever it was needed. A new era had begun." - Vestrum Tobias
The Guardian is a person in charge of controlling and channeling the Balance of science and magic between the two worlds. A Guardian of the Balance lives in a tower in the Guardian's Realm, and is replaced every one thousand years by a new Guardian. The Guardian was set up by the Sentinel at the time the Earth was separated into the twin worlds of Arcadia and Stark, one world of magic and one of science. Here the Guardian watches over the balance of magic and science between the Two Worlds, apportioning the two principles between the worlds where necessary.
The Guardian spends a thousand years in the tower separated from his/her soul so that this apportioning may be done dispassionately, and is reunited with it when the guardianship is at an end - leaving the tower to enter a world changed almost beyond comprehension in the years that have passed.
It is not known how many potential Guardians are in the worlds. April writes in her diary after speaking to Cortez that the Vanguard have been experimenting on 'potential Guardians' and essentially destroying their candidacy, though she may have been referring just to Gordon Halloway. Otherwise, it sounds as if Gordon was the only candidate to survive this experimental process, the exact nature of which is (thankfully) never revealed.
The criteria for being a Guardian are never spelled out, but they would likely involve having a personality with a reasonable balance of logic and chaos, courage to face the prospect of being essentially held in stasis for a thousand years, and a strong sense of compassion and justice. There are indications that Guardian candidates are also Shifters. Given the Three Trials a Guardian must face, he or she must also be physically strong and fit, able to use powerful magic, and psychologically stable enough to face their own personal demons. While there seems to be no discrimination of candidates on the basis of gender, it seems that only humans are eligible to be Guardian. If being a Shifter is indeed a criterion of candidature, this may support such a claim - given that it is unlikely that a magical race (who require Arcadia to exist) would develop the ability to go to Stark.
In addition to maintaining the Balance between worlds, Abnaxus tells April that the Guardian has the ability to maintain balance within worlds, and with regard to time. (This latter ability prevents the Venar from abusing their special comprehension of time.) The exact nature of these powers are not known.
We meet two Guardians in TLJ:
The first Guardian was mentioned by Vestrum Tobias Grensret; a woman who remains anonymous.
In Dreamfall, we learn a little more about an enthroned Guardian. Despite the fact the soul of the Guardian is removed, Gordon is still able to recognise and speak to April. However, his consciousness has been expanded, and he seems to be coexisting with the Balance. He also claims to be a part of, but apart from, the Guardian's Realm itself, allowing him to have control of it. While Gordon is aware of a change in the flow of dreams between the worlds, he says it does not concern the Balance, "so it does not concern me."