Examination in chief is the questioning of a party’s own witness under oath, at trial.
Witnesses are introduced to a trial by their examination-in-chief, which is when they answer questions asked by the barrister representing the party which called them to the stand.
After their examination-in-chief, the other party’s barrister can question them too which is called cross-examination. They may also be re-examined by their own side’s barrister after cross-examination.