A responsibility is something you are expected to do.
A responsibility might be a task you are expected to do. For example, your parents expect you to brush your teeth. Brushing your teeth is “a responsibility” and it is your responsibility to brush your teeth every day. Another example is that your teacher expects you to finish your homework on time and to do your best job. So it’s your responsibility to do your homework and to do it to the best of your abilities.
A responsibility might be a way you are expected to act. For example, your parents expect that if you go to play at the park, you will play in a way that won’t seriously hurt yourself or someone else. It’s your responsibility to have fun in a safe and courteous way.