The Prime Program is the title which The Architect gave to the Path of The One. In reference to computer programming, a prime program is a subprogram within a larger program (referred to as a proper program). In this case, the Matrix is the proper program, while the Path of the One is a subprogram within the Matrix.
A prime program is a subroutine that has a single entry point and a single exit point, with any number of decision nodes along the way. This subprogram was created in order to guide the One to the Source, which is the exit point for the Path of the One. This was confirmed by The Oracle, who stated that the Path of the One ends at the Source. Because reaching the Source is the exit point for the prime program, the One must reinsert it into the programming of the Matrix in order to start the cycle over again. Upon reinserting the Path of the One into the Matrix, the entry point (the first decision which the One must make) is to choose 23 individuals to rebuild Zion. Eventually the One will die, the Oracle prophesies his return to the inhabitants of Zion, and many years later the next One arises in order to complete the Path of the One and repeat the cycle.
The Path of the One is a necessary subprogram to ensure that the One will act according to the Architect's wishes. It is one of the control measures created with the help of the Oracle, who understands the human psyche. The Path of the One is designed to create within the One a messiah complex. The One is granted special powers by virtue of his connection to the Source in order to make his role as a messiah believable. Each decision node along the Path of the One is designed to create a sense of attachment between the One and humanity. Believing himself to be the savior of mankind, the One will take the necessary steps to ensure that humanity continues, albeit in an enslaved state. This process was subverted in the case of Neo, whose attachment to Trinity was stronger than his attachment to humanity. This love for Trinity was strong enough that he was willing to sacrifice humanity in order that he might save her.
The Path of the One is necessary because 1% of the population rejects the Matrix. The programming of the Matrix is not designed to handle even a single rejection. The only way for it to operate at peak efficiency is when there is 100% acceptance of the programming. The Architect's attempts resulted in 100% rejection. Yet even though the Oracle's version of the Matrix yielded nearly 99% acceptance, there was still a degree of rejection that had to be dealt with. Each rejection (referred to as an anomaly by the Architect) is in the form of code within the Matrix that acts somewhat like a "Bug". Over time, the number of rejections begins to build up and threatens the entire system. When this happens, the Architect states that it has become a systemic anomaly. This systemic anomaly must be taken care of or else the Matrix will eventually crash--killing everyone who is hardwired to the Matrix.
The solution is to take the systemic anomaly, attach it to the One and have the One bring it to the Source where it can be destroyed. This gathering of the systemic anomaly and attaching to the One is referred to as the integral anomaly. This is the code that the One carries which must be disseminated at the Source. The One stands in proxy for all those who have rejected the Matrix. By entering the Architect's chamber and choosing to return to the Source, the One has essentially accepted the Matrix and has negated all the rejections attached to him. This brings the Matrix to 100% acceptance. However, this is only temporary, as it is only a matter of time before someone rejects it. Therefore, the Architect must quickly reload the Matrix while it is still at 100% acceptance.
Reloading the Matrix is a separate step from reinserting the prime program. They can be confused as being identical because they occur at the same time. However, in the online video game The Matrix Online, it was confirmed that after the events of The Matrix Revolutions the prime program was not reinserted, even though the Matrix was reloaded. This simply means that the Path of the One no longer exists within the programming of the Matrix.
Neo, unlike the previous Ones before him, did not complete the Path of the One. Instead of going through the door on his right in the Architect's chamber, he left the Path of the One and went through the door to his left. This means that he left the Path of the One and began to follow the Path of Neo. Ironically, his own path still led him to becoming connected to the Source where he followed the original directive of the Path of the One.
Neo's unique contribution to the problem of choice was to make a deal with the Deus Ex Machina that the programming of the Matrix would now accept rejections. Those who desired to be free would be allowed to leave. This meant that when someone rejected the Matrix, it was no longer an anomaly. Therefore, with no anomalies threatening the programming of the Matrix, it was no longer necessary to have a One, which meant that the Path of the One was also no longer necessary. That is why the prime program was not reinserted. Because of the compromise made between man and machine, a state of peace was created.