Kwisatz Haderach was a term used primarily by the Bene Gesserit. It was an old Chakobsa term that translated literally as "Shortening of the Way".
Origins
The term originated from the Bene Gesserit's prescient powers, and their inability to venture into a specific region of prescient knowledge. This region, though mysterious in nature, was known to be unattainable to females. Specifically, the spice melange allowed the Bene Gesserit to unlock genetic memory, but only on the female side. For reasons unknown, they could not see into the male side, and the very thought of trying was terrifying to them. Further, melange also exposed the Sisterhood (and others) to a very limited form of prescience - thoughts, feelings, images into the very near future, but no more.
Attempts to Create the Kwisatz Haderach
The Bene Gesserit desire to uncover this knowledge and its associated powers drove them to initiate a long-running genetic breeding program that would yield a male with mental powers capable of bridging space and time, and that he would be under direct control of the Sisterhood. The Bene Gesserit knew that, not only would their Kwisatz Haderach possess Other Memory on both the male and female side, but that he would be able to predict the future precisely. He would be Bene Gesserit Reverend Mother, Mentat, and Guild Navigator, all in one being.
After the emergence of this male - Paul Atreides - the term Kwisatz Haderach was also understood as meaning "one who can be many places at once", and became synonymous with Paul. In time it also came to encompass his sister, Alia Atreides, and his son Leto Atreides II, since they all had similar abilities.
The initial Bene Gesserit plan was to breed the daughter of Duke Leto Atreides to a Harkonnen male, which would produce the Kwisatz Haderach. This was to be the culmination of more than 10,000 years of careful breeding. This plan would have seen the end of the centirues-old feud between the Great Houses Atreides and Harkonnen, and placed a prescient, Bene Gesserit-controlled male on the Golden Lion Throne, the Kwisatz Haderach. However, because of her love for Duke Leto, the Lady Jessica disobeyed her fellow Bene Gesserit and gave birth to a son instead of a daughter, to give him an heir, Paul.
It seemed fairly certain from early in Paul's life that he would in fact be the Kwisatz Haderach, since he showed an ability to see into the future. It was when he was fifteen years of age that the Bene Gesserit sent a Reverend Mother to test Paul's prescience and his training in the Bene Gesserit ways. This event appeared to be a significant catalyst for the events that would befall the universe for the next several thousand years, since Paul's testing with the Gom Jabbar, as well as the Sisterhood's silent complicity in his father's death, proved to instill significant negativity in Paul against the Sisterhood. As a result, when he reached young adulthood, and ascended to the Golden Lion Throne, he vowed that he would never be under the control of the Sisterhood. As a result, the Sisterhood lost control of their breeding program, their Kwisatz Haderach, and the possibility of placing a Bene gesserit on the throne.
The Bene Gesserit Sisterhood were not the only ones to manipulate genes to produce a Kwisatz Haderach. The Bene Tleilax also managed to breed their own Kwisatz Haderach, some time during or prior to the rule of Paul Atreides. However, faced with being trapped by his own destiny, the Tleilaxu Kwisatz Haderach chose to kill himself. Indeed, it appeared that the majority of people did not realise that to know the future is to be trapped by it. Paul could see that he would have to lead humanity onto a drastic course in order for it to escape its own annihilation. As a result, he chose to escape the Golden Path by allowing himself to be blinded, so that he could walk into the desert and not be a burden on his tribe. The mantle of the Golden Path was subsequently taken up by Leto II.
Kwisatz Haderach's Impact
As Kwisatz Haderach, both Paul and his son Leto would at times experience significant grief because they knew they were trapped by their own destinies. One of the few people who understood this was in fact one of Paul's enemies. It was through the Tleilaxu Kwisatz Haderach 'experiment' that the Tleilaxu Master Scytale could in fact sympathise with Paul and the decisions he had to make.
When Leto II, the last Kwisatz Haderach, ascended to the throne, not only did he manage to avoid Bene Gesserit control, but he also took control of their breeding program, and relegated them to a relatively insignificant role in the universe throughout the 3500 years of his reign.
During the Scattering and Return
After the Scattering promoted by Leto II after his death, the remaining power brokers in the Old Empire either forgot about the concept of the Kwisatz or vehemently guarded against the rise of another of the super-beings. The Bene Gesserit would go so far as to terminate the life of humans that showed aspects of enhanced abilities.