What was the system under which individual families contracted with landowners to work a plot of land in return for a portion of the resulting crop is know as?

The system you are referring to is known as sharecropping. In this arrangement, individual families, often freed African Americans in the post-Civil War South, would farm a plot of land owned by landowners. In exchange for their labor, the sharecroppers would receive a share of the crops produced, which they would use to pay off debts for supplies and rent, often leading to a cycle of poverty and dependency.