The adversial system, or adversary system as it's also know, is the system which settles disputes by one party lodging a complaint against another, a la plaintiff (person who brings forth the complaint) versus defendant (person who the complaint is being brought forth against).
Australia uses the adverserial system - in civil cases, it is one party vs. another (e.g. Smith vs. Cook) and in criminal cases it is one party vs. the state/commonwealth (e.g. Smith vs. NSW).
All cases held in Australian courts are adversial, and even arbitration, etc. could be considered adversial.
Many, however, disapprove of the system because it does not seek to find the truth per se - more often, lawyers and barristers can spin webs of lies to hide the truth behind mountains of red tape.