What does it mean to have a “civic duty”?

A civic duty is something that a person does out of a sense of responsibility to society. It may or may not be punishable by law. For example, serving on a jury is part of civic duty. While it is illegal to ignore a jury summons, many people try to get out of it. By serving, one shows one's willingness to ensure that society functions. Voting is another civic duty. Legally, one is not punished if one does not vote; however, by voting, one demonstrates that he or she is willing to take part in a world outside of oneself and be an active participant in democracy.
The concept of civic duty goes back to an idea of "virtue" which was held during the Enlightenment. Having "virtue" meant that one took part in the civic life of a community without the expectation of a direct and immediate reward. One does not personally get anything for voting, but one's participation ensures that the government can hopefully best protect the rights of all.