If you have responsibility for something or someone, or if they are your responsibility, it is your job or duty to deal with them and to take decisions relating to them.

If you accept responsibility for something that has happened, you agree that you were to blame for it or you caused it.

Your responsibilities are the duties that you have because of your job or position.

If someone is given responsibility, they are given the right or opportunity to make important decisions or to take action without having to get permission from anyone else.

If you think that you have a responsibility to do something, you feel that you ought to do it because it is morally right to do it.

If you think that you have a responsibility to someone, you feel that it is your duty to take action that will protect their interests.