n-th generation head of a family
Suffix attached to a number or generation counter to indicate the ordinal head of a family or clan. Used in historical or formal contexts.
He protected the family as the fifth-generation head.
The third-generation head of this lineage was a famous scholar.
代 is a suffix meaning 'generation' or 'era'. 代当主 combines it with 当主 to specifically mean 'head of the family of a certain generation'.
Compound of 代 (だい, 'generation') + 当主 (とうしゅ, 'head of a family').