The Marduk Institute (マルドゥック機関, Marudukku Kikan) is, supposedly, the secret selection committee that screens the global population for 14 year old children who would make suitable Evangelion pilots.
However, Ryoji Kaji's investigations in Episode 15 reveal that the Marduk Institute does not really exist; it is a dummy organization composed of 108 front companies, and Eva pilots are actually hand-picked by NERV. All potential Eva pilots are then transfered to Class 2-A in Tokyo-3, to keep them safe and on-hand should NERV require them.
The factors for selection are unclear, as is the selection process itself. Whether Ritsuko alone picks the candidates (with or without consulting Gendo), or if Gendo and Ritsuko are involved together remains unanswered. Since Fuyutsuki and Chairman Keel are listed on the board of directors of the phantom companies, they might have a (limited) degree of involvement as well, or at least know the truth about the Marduk Institute.
Analysis of Gendo's conversation with Ritsuko
Following Shinji running away from home in Episode 04, Gendo tells Ritsuko that the Fourth Child has yet to be found as per the Marduk Institute's report. Aside from the fact that this reveals that there is a Third Child who has yet to appear in the story, Asuka and Unit-02, the conversation seems meaningless. Gendo and Ritsuko both know that Marduk is a sham and NERV picks the pilots itself. Gendo and Ritsuko (with Rei following them) are essentially in private and don't need to make the conversation up. Thus it is likely true that Toji has yet to be selected, and they must be mentioning the Marduk fiction simply because Rei is present.
This still leaves at least one question: When Toji Suzuhara is selected in Episode 17, Ritsuko picks him because a core can be configured for him. Shinji synchronizes with Unit-01 because his mother Yui's soul is in it, and Asuka synchronizes with Unit-02 because her mother Kyoko's soul is in it. The Classified Information makes it clear that the Pilot/Mother relationship is true for all Evas with the exception of Unit-00, thus Unit-03 contains the soul of Toji's mother. The timespan of Evangelion construction appears to be over a span of years in case of earlier Evangelions and only a few months for the Mass Production Evangelions. Whether they select the most suitable pilot and then use the corresponding soul, or vice-versa remains unanswered.
- The "Marduk Institute" is really 108 front companies: this might be yet another religious reference, as there are 108 sins in Tibetan Buddhism, and 108 beads on a Hindu or Buddhist rosary, with a separate head bead.
- In Buddhism, 108 is also the number of steps or earthly temptations that a man must overcome to attain Nirvana.
- In Hinduism, several powerful Gods and Goddesses have 108 names. In this case, all 108 different names are merely aliases for a single organization: NERV itself. The Babylonian God Marduk, after whom the organization is named, had fifty names himself.
- In Babylonian Mythology, Marduk was the God who created humans to bear the burdens of life so that the Gods could be at leisure.