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The one-drop rule is a term referring to racist rules in the Jim Crow laws that classified as black anyone with any African ancestry. These laws were all passed in the era between 1910 (when Tennessee passed the first one) and 1931 (in Oklahoma). Regardless of one's appearance, any African parentage meant you were legally considered a black person for the purposes of segregation. Similar rules applied to Native American ancestry, though less strictly as many prominent families claimed descent from Pocahontas.
In non-American usage, this concept is referred to as hypodescent, or the idea that any mixing of social classes means the progeny belong to the lower class.
In previous eras, many prominent lawmakers had pushed back against this type of law, knowing well that nearly every prominent family had some African ancestry, but as this memory retreated, family myths erased such fears.
While no longer a current legal concept, the underlying attitude is still prevalent is American culture. For example Barack Obama is widely considered to be, and self-identifies as, African-American (or "black"), even though he has equal African and European ancestry. Colin Powell's ancestry includes "English Irish, Scotch, probably Arawa Indian, and a Jewish strain" and even has a Scottish coat of arms. This attitude is different from South Africa for example, where people of mixed ancestry are considered coloureds and not black, which is used for people of primary African heritage.