A family commitment- An appointment with family-or going to do something with family. For example- My friend invites me to dinner, i might reply, I'm sorry I have a family commitment(my daughters school play is tonight) Lets go to the park!- Sorry I have a family commitment-~(I have to go for lunch with my mum) Household chores/Looking after kids/Earning money to support family-These are not family commitments-they are responsibilities or duties you have to your family. Here commitment means -a promise (to do something).