Among the most popular: paternity and kinship testing, which adopted children can use to find their biological relatives and (which) families can use to track down kids put up.

Among the most popular: paternity and kinship testing 冒号代替了系词is。这是一个倒装句。= Paternity and kinship testing is among the most popular.

例如：The president of the company: Mr. Smith. = The president of the company is Mr. Smith.

which... and (which)为并列的非限制性定语从句，修饰句子的主语 paternity and kinship testing。关系代词在从句中作use的宾语。由于主语带了很长的定语从句，所以本句主语和表语互换位置（主语后移，表语前置），形成完全倒装句。

译文：其中最受欢迎的检测之一是父子和血缘关系检测，被收养的孩子可以利用它找到自己的血亲，家庭成员可以用它来追踪到被收养的孩子。