Grigori was a government organization that studied PSI abilities by performing inhuman experiments on children. The organization's first plan was destroyed by test subject #01 when he escaped 16 years before the series, however, it was rebuilt and continued to perform PSI experiments. Grigori's second plan was more sadistic and unconcerned about their methods, all in order to prevent another failure. They made the test subjects wear uncomfortable helmets that suppressed their PSI, implanted chips in the test subject's brains that can incapacitate them, strengthened their own minds to make themselves immune to Trance, and brought Iba to take care of the test subjects and make sure they didn't rebel against them. Grigori was finally destroyed one year before the start of the series when Iba turned off #06's helmet, who used his immense powers to kill everyone but Iba.
The director of Grigori treated the children as test subjects and never saw them as humans; he also manipulated #06 by promising him freedom for him and his sister so that they could live with their parents and even made fake letters to convince him to go along with the experiments.
Kouichi Iba was the lowest ranked member of Grigori and was responsible for the care and psychological counselling of the children. Iba spent a lot of time with #06 and even gave him a birth date. By the time #06 realized the lies that the director told him, Iba was the only one who communicated with him.
The name of the organization may be a reference to the "Grigori," or "Watchers" from the Apocryphal Book of Enoch. Though some are benevolent, most are Fallen Angels who were condemned to live on Earth for giving humanity the secrets of God. In the bible, the proliferation of their half-human offspring was the main reason God ordered the Great Flood, in order to wipe the world clean and begin again.