The Architect created the first Matrix as a utopia for the humans whose minds inhabited it. However, the human minds rejected this first attempt as a perfect world and beta 1 of the Matrix crashed. A second attempt added "vulgarities" of human nature and a basic cause and effect, but this beta was also a failure.
The Architect turned to a more intuitive program designed to understand human nature and psychology to augment the framework of the next Matrix. This time, the power of choice was added to the programming, where humans would be allowed the power to choose, even if the person was only aware of the choice on a vague, subconscious level.
The Architect noted that the Matrix was not as perfect as he would have wanted. The choice-programming that stabilized the first iteration of the Matrix was also causing its destabilization. In approximately 100 years time, a reload of the Matrix would be required.
There was also the matter of Zion, which could not be allowed to grow too large with rejected humans, lest its people would rise up to become a new threat to the Machine City.
One significant anomaly appeared two-thirds into the Matrix first cycle. A man was born with control of additional Matrix programming that actually belonged with Source. The Architect tried to remove this anomaly, to no avail. However, he would find a way to work with this human and return the critical Source code he carried for the eventual reloading of the Matrix.
Together with the human intuitive program (which could be considered the "mother" of the Matrix as the Architect could be considered the "father"), the concept of the Prophecy was formed. The intuitive program (known to the humans as the Oracle) would tell of this story to the small members of a human resistance that periodically infiltrated the Matrix, who would find the anomaly and help him to find the Architect's hidden room deep within a fortified room. There, the Architect would use his measures of control to keep the Anomaly, and in turn both Zion and the Matrix, in check.
In each of the first five cycles of the Matrix, the Anomaly, known to the humans as The One, would manifest itself within the Matrix and eventually find the Architect and his hidden lair. There, the Architect (who is also bound by the choice-programming within his own creation) presents The One with two doors of exit. One door leads to the Source. The second door returns to the Matrix proper. The Architect tells the One that, while they converse, Zion is about to be utterly and totally destroyed. He states that the function of The One is to save humanity. Given that the only remaining humans (after Zion's destruction) remain in the Matrix, failure to enter the door to the Source will lead to an eventual catastrophic system crash of the Matrix, killing its human inhabitants and effectively annihilating humanity as a race. The Architect offers to The One a token number of humans who he can use to rebuild Zion. In each of these cycles, The One enters the door to the Source, the Matrix is reloaded, and Zion dies and is reborn at The One's mentorship.
On the sixth iteration, Neo, the sixth Anomaly, appears on schedule before the Architect. The Architect is surprised that this One, unlike his predecessors, is quicker of thought. This sixth Anomaly possesses the same dispensation for protecting humanity as the others, but unlike the other Ones has a deep attachment to one human: a Zion resistance member named Trinity.
The Architect delivers the usual speech and threat, but can already see that this One will not comply as the others. Neo leaves the Architect to save his love, and leaves the future of the Matrix in doubt.
The Oracle tells Neo more about the Architect at their final meeting. She says that the Architect's purpose is to balance the equations of the Matrix, but her purpose is to unbalance the same equations. She tells Neo to head to the true location of the Source, the Machine City, to save not only humanity, but the Machine world as well.
After Neo's destruction of Smith, the Machines gather Neo's body and successfully reload the Matrix. The Machines also oblige the truce that Neo bartered, calling off the attack of Zion and leaving the two worlds at peace.
The Architect meets the Oracle and speaks of her "dangerous game" that she played, and acknowledges that the truce will be honored, although he doubts how long it will last.
The architect's appearance is likely a homage to Vinton Cerf, who is one the creators of the internet.