Individuals capable of piloting the Evangelion who are selected by the Marduk Institute, an advisory body under the direct control of the Instrumentality Committee (the Marduke agency is in fact NERV). They are called First, Second, Third, etc., according to the order of their selection. What is common to these Children is that they are all young boys and girls who have lost their mothers. Incidentally, the candidates for Children were grouped into Class 2-A of the First Municipal Junior High School of Tokyo-3.

Source: The End of Evangelion : Glossary contained within The End of Evangelion - Theatrical Program.

The use of “Children” in the titles the First Children, Second Children, Third Children, is not a mistake. It should not be First Child, Second Child, Third Child; the use of “Child” is incorrect (no matter what AD Vision tries to tell you). GAINAX chose to use the English word “Children” as the identifier of the Evangelion pilots, it is not a translation mistake, it was intentional. In Neon Genesis Evangelion the terminology used to describe the pilots is written "tekikakusha" which translates to "qualified/adequate person", however it is pronounced "Children". It was an intentional choice to use Children, and as the series progresses the reason behind that choice becomes clear.