Actually, it's a binding arbitration

All parties to the suits on the show signs a contract to accept Judge Judy's ruling.  It is called binding arbitration.  

Arbitration (仲裁）is a form of alternative dispute resolution that functions like a court, with a arbitor and two parties, but it has less evidenciary requirement and less procedural protection, overall reducing the cost of litigation.  

The spectrum of alternative dispute resultion, from most costly to less costly, are generally:

lawsuit, arbitration, mediation, fact-finding, and negotiation.  There are different formats within each category.