Family members (spouses, children under the age of 21) will receive the H-4 visa. In this category, family members are not allowed to engage in any kind of employment (to be allowed to work they need a work visa, e.g E-1, E-2 or L-visa). Children may of course attend school or a college/university.
IMPORTANT: unmarried or same-sex partners do not receive a related status, the latter even if they are married under foreign law or US state law. They will have to apply for their own (work) visa.