The Head Office or Emergency quota is a handful of berths per train that the zonal headquarters controls. A passenger with a waitlisted ticket can apply for it through an Emergency Quota form at the divisional or zonal office, usually with a reason such as a medical need, a bereavement or official duty, and the allotment is decided at charting.
It is not a booking channel and it is not for the general public in practice: the quota is tiny, the request must reach the right office in time, and many stations do not accept applications from unconnected passengers at all. Most people who mention it are describing a request made through an MP, MLA or railway officer.
Also written or searched as: emergency quota · EQ · HO quota meaning · head office quota