You need to file USCIS Form I-612 if you, having lived in the US on a J-1 visa, seek a waiver based on (i) legitimate fear of persecution if you return home or (ii) exceptional hardship to your US citizen spouse or child.
Form I-612 is also required for waiver applicants who are spouses of J-1 visa holders (in J-2 status) who are no longer married to the J-1 spouse at the time of the waiver application; or children of the J-1 or J-2 visa holder who are themselves married or who are over 21 years of age. As with the J-1 waiver application, Form I-612 is only needed if the basis for the waiver is fear of persecution or exceptional hardship (to the J-2 applicant, not the J-1 visa holder).
J-2 spouses who are still married to the J-1 holder, and unmarried children of the J-1 holder who are under 21 need not file Form I-612, as long as you file Form I-612 and include their names and information in your application. If you fail to do so, they will have to file their own Form I-612 to remain with you in the US.