Answer by Chris Peters:
The Architect is the creator of the Matrix. He sits isolated in a room of monitors displaying everything that happens, and simulations also help the Architect predict everything that could happen. Thus he is omniscient, all knowing, about events in the Matrix.
His strange language hints at a mind obsessed with perfection. The Oracle claims the Architect can only understand the world mathematically ("an equation to be balanced") and cannot see beyond this way of thinking. The Architect brags about the perfection of his designs even when they fail. As he states in Revolutions, he does not go against his word. Thus the Architect means exactly what he says, no matter how strangely he says it.
This is the 6th version of the Matrix. The first Matrix resembled the Garden of Eden, and the second simulated some unspecified period of human history. Versions 1 & 2 failed "monumentally" until the Oracle discovered a solution for version 3.
The Matrix has a "systemic anomaly" – human Free Will. Individual choice is an imperfection in a computer system, where everything must operate according to set rules. And yet without Choice, versions 1 and 2 failed. The Oracle introduced choice on an near subconscious level – choice to accept the Matrix or not. Most humans, 99%, chose to accept it. The nature of Free Will is explored throughout the sequels, with almost every major character debating Neo on the meaning of choice.
The One is the "integral anomaly" or ultimate expression of choice. Like the Matrix, Neo is the 6th version of The One. While Zion represents free will across the general population, the One is the embodiment of Free Will – he (or she?) questions everything to the point that he can rewrite the Matrix code at will. Both represent a threat to the Machines.
"The One" is also a system of control to limit the Anomaly. The Machines monitor the One closely throughout his life, inserting code into his being, all with the purpose of leading him to the Architect's room. He is given a chance to repopulate Zion, allowing the Machines know where Zion is and how big its population is. By eliminating Zion, the Machines also limit The One's base of support and gain his voluntary cooperation.
The choice offered by the Architect is flawed. The Architect gives Neo a choice of sacrificing a few (Zion) to save many (The Matrix). This is a variation on Philippa Foot's, a similar moral dilemma where most people choose to save the many. Neo, however, recognizes there are more possibilities than the Architect is letting on. As the Oracle has predicted, Neo has had visions of Trinity's impending doom and has already made the choice to save her and "sacrifice" the Matrix. Thus he sets in motion outcomes that no one has predicted.