The exchange visitor/training J visa is provided for persons who are approved to participate in exchange visitor programs in the U.S., under provisions of U.S. immigration law and the State Department. This means that before you can apply at a U.S. Embassy or Consulate for a J visa, you must first apply, meet the requirements, and be accepted for one of the Exchange Visitor Program categories through a designated sponsoring organization. If you are accepted as a participant in an exchange program, the sponsor will provide you with information and documents necessary to apply for the J visa to enter the U.S.
If you are authorized by your sponsor to be accompanied by your spouse (and children, they can apply at the same time.
Ask your sponsor if they have a training program in place for the maximum 18 month stay. There are no extensions allowed for J visas and some countries require you to return to your home country for a minimum of 2 years following a training program. Check the State Dept. website for further details.