A responsibility is something you're required to do as an upstanding member of a community. If you're in a band, it's your responsibility to learn the songs and show up to practice on time.
Responsibility comes from the Latin responsus, which means “to respond.” There are actually a few different definitions of the noun. It can be another word for trustworthiness, as in, "He demonstrated his responsibility by showing up to practice on time." And it can be used to describe the social force that motivates us to take on individual responsibilities, as in, "A sense of responsibility drove him to memorize all of the songs."