What does it mean to act responsibly?
I am assuming that the question, in some aspect, refers to a character acting responsibly in a text (since it has been posted under literature). Acting responsibly means that one acts upon his or her duty because of obligation, a role in life, a job, or based upon one's accountability to another.
In regards to an example in literature, one can find a character acting responsibly in Harper Lee's To Kill a Mockingbird. Atticus Finch, the father of Scout and Jem, is asked to take the case of an African American man (Tom Robinson) accused of rape by a white woman (Mayella Ewell). Knowing that Tom deserves a fair trial (or as fair as possible), Atticus agrees to represent him--even though it goes against how the majority of the town feels. Atticus understands his responsibility to mankind and humanity. Therefore, he acts responsibility by taking Tom Robinson's case.
As for an example of not acting responsibly, this can be seen in John Steinbeck's Of Mice and Men. Lennie Smalls is very simple man (in regards to needs and mental ability). Although George knows this, he still allows Lennie to take one of Slim's puppies. Lennie has killed many animals in the past (he simply does not recognize his own strength). By allowing Lennie to care for another animal, George is not acting responsibly. Lennie has killed in the past, and George knows that Lennie will (most likely) kill again.