After the recovery of Unit-01 by Asuka, and the extraction of Shinji from the Core, the collar was fitted around his neck as a result of his actions which led to the Near Third Impact. Its primary function is to kill him if he loses control of his emotions again leading to the awakening of an Eva as it did during his fight with the Tenth Angel.
This necklace is connected by a coded detonator that Captain Katsuragi holds.
Kaworu Nagisa is later able to remove it from Shinji's neck and place it on his own in order to gain Shinji's trust. During the Fourth Impact, the collar activates around Kaworu's neck, generating some sort of floating cubes that converge upon his neck, thus killing him.
Asuka wears one as well most likely to kill her if the Ninth Angel were to take control.