What's a "gateway operation"?
Looking at this FAA notice from the 2012 RNC convention in Florida, there are multiple references to "gateway operations" and "gateway airport(s)". Specifically:


  Aircraft not operating under a TSA-approved standard security program ... when the TFR is in effect must be security screened by TSA at a designated gateway airport or location.
  ...
  TSA security screening and gateway operations will be available on August 26 for those operators wishing to operate into St. Petersburg-Clearwater International Airport.
  ...
  TSA security screening at the gateway locations will consist of ID 
  verification of all passengers and crew, watchlist matching, physical screening of persons and baggage, and inspection of the aircraft.
  ...
  Positive pilot identification procedures will be in effect during this TFR. Special ATC procedures and routes to and from this airport may be assigned prior to departure from a gateway airport. Upon departing the gateway airport, all aircraft must be on a VFR or IFR flight plan, maintain radio contact with ATC and continuously squawk an ATC-assigned discrete transponder code. Intermediate stops are not authorized unless there is an emergency. If an intermediate stop occurs, aircraft will be required to return to a gateway airport to be re-screened prior to continuing flight with the TFR


Then referencing this TSA post:


  Dulles International Airport (IAD) will serve as the newest gateway airport, allowing certain pre-cleared general aviation operations, including corporate aircraft, charter flights, and on-demand operations, to fly directly into Ronald Reagan Washington National Airport (DCA).
  ...
  Aircraft must arrive from one of 17 gateway airports after flight crews have been vetted and passengers screened.


So it would appear that "Gateway operations" are specific security protocols for departing and arriving aircraft within a certain restricted area.

EDIT:

It appears that, if "Gateway Airports" and "Gateway Operations" are synonymous, that it is about strict preclearance procedures defined post 9/11. As far as I can tell, these restrictions are permanently in place for flights in Ronald Reagan National Airport (DCA), due to the obvious risk the airport's proximity poses to various Federal government buildings. 

These rules are called "DASSP" for DCA Access Standard Security Program. The consist of the following regulations


  TSA inspection of crew and passengers.
      TSA inspection of property (accessible and checked) and aircraft.
      The start and end dates of flight must be indicated on the TSA flight authorization.
      Identification checks of passengers by TSA.
      Submission of passenger and crew manifests 24 hours in advance of flight.
      Enhanced background checks for all passengers and fingerprint based criminal history records check (CHRC) for flight crew.
      Armed Security Officer (ASO) on board each flight.
      All operations are subject to cancellation at any time.
      All unscheduled operations to/from DCA require an FAA Slot Reservation.