A red spherical organ common to all the Angels, and believed to be of critical importance to their construction. Some early Angels had this organ somewhat exposed, allowing the Evangelion Units to assault it directly. However later ones came to employ defensive measures to protect it (such as Zeruel, the Fourteenth Angel).
It is confirmed that Evangelion Unit-00, Evangelion Unit-01, and Evangelion Units
Because the Sē Engine plays such a vital role in the series but has never been shown on-screen, there is speculation that the Core may be the Sē Engine which supplies energy to the Evangelion Units and the Angels. This theory is not widely accepted however, as Evangelion Unit-01, which has been shown to have a Core in its chest cavity throughout the series, does not initially possess Sē Engine functionality (thus having to be powered by an Umbilical Cable).
A more consistent version of this theory suggests that the Evangelion Cores are simply incomplete, possibly due to technological constraints at the time of construction.
This may also explain how the Evangelion Units can go berserk even without power, by being able to temporarily activate the Core to its full potential (albeit only under extreme circumstances). In this case, when Evangelion Unit-01 consumes the remains of Zeruel, the Fourteenth Angel, it may have only consumed vital genetic information which allows it to supplement and complete its own Core, thus granting it the full functionality of an Sē Engine.