If you sign contracts for goods and services in your business, you should be aware of mediation as a dispute management process. BEFORE you sign that contract, you should be investigating the legal considerations of that contract, and this is most often best done by discussing it with your own lawyer.
It is during these discussions and during negotions with the other party or parties to your contract that a provision be inserted to the effect that the parties agree that should any dispute arise, the parties will attempt mediation before pursuing any other legal remedy; ie, arbitration or litigation (both of which are third party settlement options, meaning you give up your right to be part of any solution or settlement in favor of a third party, who will impose settlement upon the parties).
Lawyer/mediators too often are adversarial in their approach to disputes (that is what they are trained to do, after all), and you may find mediation discussions dragging on, and direct exchanges limited by rules and regulations.
Good mediators must have significant experience in settling claims in legal forums, and must understand the law. Most belong to professional groups, which bind them to ethical standards of behavior, and they should be willing to be interviewed by the parties involved in any contract dispute before any hearings begin.
Parties must understand that mediation can and often does occur even as disputes move through the courts or through arbitration proceedings, and if mediation helps the parties find a solution, well that's what the saying "they settled on the courthouse steps" means, and that is, in fact where more than 90% of litigated disputes end up settling.
Ask around, and don't be afraid to call and talk with potential mediators. If they are experienced, they can orchestrate such discussions with the parties withou in any way prejudicing potential settlement options. One last point: if you don't know by the end of one day whether a settlement is possible, you have likely selected the wrong mediator to help your process along, and your mediator is probably also an attorney.