Ritual is defined by sociologists as "that part of family life that the family likes about itself, is proud of, and wants formally to continue."

that 是“指示代词”（那一），作定语，限定 part。

formally 是“副词”（正规地），修饰不定式 continue，不是修饰wants。一般而言，修饰不定式的副词多在 to 之前。

that the family likes about itself, is proud of, and wants formally to continue，是定语从句，修饰 that part of family life。定语从句含3个并列谓语：

likes

is proud of

wants