종갓집 (or just 종가) means the family or home of the eldest son of the siblings (종 = primary, head, 가 = family, home, ㅅ = 사이시옷 for compound words, 집 = family, home).
Traditionally, the eldest son is considered to carry on the family lineage, and so has special duties and authority in family matters, until he passes away and his own eldest son takes up the same role. This shows Koreans traditionally have placed great importance on sustaining the family blood, with male children and especially the eldest male child playing the central role.
This tradition is however no longer observed as much nowadays and families don't feel they must have at least one male child as in the old days. So the significance of 종갓집 is more and more becoming a thing of the past.