*Deep Breath*
Basically, the Architect is the program that created the Matrix, and appears in human form to Neo.
He explains to Neo that his powers are a result of the way the Matrix is made, and that it isn't the first iteration of the Matrix, since it's a cyclical thing - it keeps being restarted.
Originally, they had everyone connected to the Matrix, and it was a failure because people kept dying as a result of rejecting the Matrix (or something like that, I can't remember exactly.)
To remedy this, they reconstructed the Matrix, and allowed for people to reject the Matrix and leave. The rejection only happened to a small number of people, and this wasn't a problem for the robot population.
Unfortunately, this new Matrix resulted in "The One" being created every time they reset it. As well as having some cool powers, The One carries some kind of code with him, which is necessary to reset the Matrix, and start again.
The Architect gives Neo a choice - return the coding he carries with him to the source (like a good boy.) The only problem is, is that Zion will be destroyed, although Neo will be allowed to choose people to repopulate Zion, so that the people who reject the Matrix have somewhere to go.
The alternative to that option is to go back into the Matrix and save Trinity, who's suffering from a gunshot wound, having jumped out of a building. If he does this, the robots will undoubtedly obliterate Zion, and kill everyone.
Neo chooses to save Trinity, believing that he thinks the humans can win against the robots this time.