Arbitration is a method of resolving legal disputes that does not involve a court. Typically the parties involved in the dispute select an arbitrator to come to a conclusion about their dispute and agree to be bound by the decision.
The Treaty of Armens provided for arbitration as a means of settling disputes between its parties, the United Federation of Planets and the Sheliak Corporate. The Treaty stated that the petitioner had the right to select the arbitrator who would hear the dispute. (TNG: "The Ensigns of Command")
Arbitration was used on Ventax II. The only example of arbitration involving an alien party prior to 2367 was a dispute between a Klingon craftsman and the owner of a home over services rendered during construction. The two parties selected an arbitrator who then resolved the matter. They felt that this would be better than using the courts due to the involvement of an alien.
That example was used as a precedent by Captain Jean-Luc Picard in 2367 during a dispute with a person claiming to be Ardra, a figure from Ventaxian mythology. Ardra claimed that she had arrived on the planet to enslave the population per the terms of the Contract of Ardra. Picard, suspecting that the woman was a con artist, challenged the validity of the contract and requested an arbitration. He, along with the impostor, agreed to have Data serve as an arbitrator, as he would not be affected by emotion or bias. The proceedings did expose the impostor as a con artist, and resulted in the dissolution of the contract. (TNG: "Devil's Due")