How do terms apron, ramp, tarmac, taxiway, parking area, gate, hardstand relate to each other?
Apron and ramp (in US) mean the same, though the official term is apron. FAA Advisory Circular Number 120-57A Surface Movement and Guidance Control System defines it as:


  Apron (Ramp). A defined area on an airport intended to accommodate aircraft for purposes of loading or unloading passengers or cargo, refueling, parking, or maintenance.


Now, the apron includes parking, maintenance and service areas, including taxi lanes.


  The apron area includes the following components: 
  
  (1) Aircraft Parking Positions.  Intended for parking aircraft to enplane/deplane passengers, load or unload cargo.
  
  (2)  Aircraft Service Areas.  in or adjacent to an aircraft parking position.Intended for use by personnel/equipment for servicing aircraft and staging of equipment to facilitate loading and unloading of aircraft.
  
  (3)  Taxilanes.  Apron areas which provide taxiing aircraft access to and from parking positions.
  
  (4)  Vehicle Roadways Markings.  Identified rights of way on the apron area designated for service and Aircraft Rescue and Fire Fighting (ARFF) vehicles.


So,


  Is this a binary split: Runways vs tarmac?


Kinda. Basically, the areas in airport where aircraft are involved are divided (by the FAA atleast) into runway, taxiway and apron. 


  i.  Movement Area.  Refers to the runways, taxiways, and other areas of an airport which are used for taxiing or hover taxiing, air taxiing, takeoff, and landing of aircraft,exclusive of loading ramps and aircraft parking areas.
  
  j.  Non-movement Area.  Refers to taxiways and apron areas that are not under ATC
  
  Is the gate stand part of the tarmac?


Its part of ramp, so yea.


  Is a parking area different from a ramp?


It is part of the ramp/apron


  Is a taxiway considered part of the apron?


Nope. As already noted.

FAA AC No: AC 150/5300- 13A Airport Design defines runways and taxiways as:


  Runway (RW).  A defined rectangular surface on an airport prepared or suitable for the landing or takeoff of aircraft.
  
  Taxiway (TW).  A defined path established for the taxiing of aircraft from one part of an airport to another. 
  
  Are aprons made of Tarmac?


Tarmac is a trademark of Tarmac limited, which supplies construction materials for a a number of airports (among others), though most of the aprons are actually made of concrete.


  Is a de-icing area part of the ramp?


As de-icing is part of servicing, it should be part of apron/ramp.

However, it should be noted that though this is the 'official' version, differing local usage means that tarmac is used interchangeably with apron/ramp.