Emergency quota or headquarter quota are limited no. of berths available to the high ranking official of railways, ministers etc.
Purpose of emergency quota:
This quota is there to provide for train tickets in emergency like situation or for an urgent surprise inspection the officer might need to conduct. Depends on situation.
Process for applying for emergency quota.
You...
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need to have contacts/friends with people in railways.
(note. They don't ask for money for this,
they shouldn't)
Then you make a request to the official.. Then that official sends a formal request (in a prescribed format) to the department which handles this.(which takes time to get processed)
Generally a request is sent in working days I.e monday to Friday, between 9 am to 5 pm (the earlier you send request the better) , getting quota done on Saturday or Sunday is extremely difficult. Chances get reduced due to the fact that nobody would be in the office to process the request( on Saturday or Sunday) , if somehow sent to the department.
How is quota alloted?
The quota or berths alloted on the basis of the seniority of officials who have sent request to the department for the concerned train and class (1AC, 2AC, 3AC.....etc)
Few ground rules for seeking a quota.
1. If the train is before 5 PM then quota is released (chart is prepared) a day before.
2.If the train is after 5 PM, then quota is released that day itself.
What are the chances of getting a berth?
Well it depends on few factors:
1.No. of people who have sent request and which ones are senior to that official.
2. No. of quota berth are there in a class. (generally 3AC have more no. of HQ quota berth than 2AC and in 1AC very few berths)
Chances: 50-50 or getting the berth and not getting the berth.(during non-peak seasons)
70 % of not getting 30% chance of getting in peak season.

Source - click here

@Roshan Sasidharan*: Re# 140389-0
Dear Roshan Sasidharan, HO quota is available for any one who can prove the urgency of the travel. But a fixed protocol is followed. The protocol starts from the President of India and down to Last Grade Government Servant and also includes other VIPs such as doctors, etc including common citizen. Since availability is limited, the allotment is based on first cum first served and the protocol ranking. The discretion of the Chief Reservation Supervisor and the staff in charge of chart finalisation plays an important role. One has to apply for release of HO quota and submit the application to the CRS on the morning of the date of journey. ...
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For the trains for which charts are prepared, the application has to be given on the previous day itself. All such applications would be grouped together and sorted on the priorities mentioned above and informed to the Divisional office concerned. One can also send their application to the Divisional office also direct either in person or through fax. If one has some good contacts within the railway division, his chances of getting included in the priority list is more.
I have been successful in getting the HO quota release a couple of times for my journeys from cbe to mas and once from mas to ndls because of my official position in state government.
We are generally always successful in getting the HO released for the travel of IAS officers. It seems that a few seats in the trains connecting the district HQ with the state HQ are allotted to the district administration concerned - DC/DM office, which we could be released for senior Government officials.

Other Quotas: There are several kinds of quotas for tickets for various special categories of passengers, other than the quotas for intermediate stations mentioned above, ticket agencies in other towns ('OS' = Out-Station quota), etc. These include small numbers of seats and berths set aside for railway officials ('HO' = Head Office quota ('High Official' quota)), ministers and high bureaucrats as well as members of parliament and their staff ('PH' quota ('Parliament House')), defence officials ('DF' quota), foreign tourists ('FT' = Foreign Tourist), handicapped travellers ('HP' = Handicapped Person), railway staff on duty ('DP' = Duty Pass) (this has its own waiting list, the 'DPWL' or Duty Pass Waiting List, especially for railway staff travelling on Privilege Passes on Rajdhani trains), women travellers ('LD' or 'LQ' = Ladies Quota), and those needing to travel for extremely urgent reasons ('EQ' = Emergency Quota), etc. Each one of these quotas can theoretically also...
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have its own waiting list, usually denoted by the quota abbreviation followed by 'WL', e.g., FTWL, EQWL, HPWL, etc. RAC or Reservation Against Cancellation (see above) is technically not a quota, but is often shown as one, as 'RC' quota.
Source irfca.org/faq/faq-travel.html