What does it mean to take responsibility for taking care of someone? What does it mean to take responsibility for someone?

It can mean many things. Sometimes it means parents having responsibility for their children (housing, food, clothing, education, etc.). It might mean having power of attorney to make decisions on behalf of someone incapable of making his or her own decisions. It might mean taking in an elderly relative (a parent?) and providing housing, food, supervision, personal care, etc. It can mean lots of things, depending on the situation.

It might also mean a parent or guardian taking responsibility for damage caused by someone in their care, as in paying for damage to a building (graffiti, broken window caused by a stray baseball or golf ball, etc.), or other such damage.