That which a citizen
is expected to do, in order to be properly described as a "citizen".
Hrm...pseudo-recursive definitions aside, a civic duty is something that we do primarily to benefit the nation. "Nation" here is a bit of a fungible term. Come to think of it, so it "duty". If you were to ask citizens of France what their civic duties might be, you would get a very different list than that of U.S. citizens. If you asked the same question of two U.S. citizens, you would still get different lists.
To get more specific:
I harken back to my AP U.S. government class lectures, where the topic of voter turnout came up. If you look at the percentages of French voters that actually vote, they are much higher than those in the U.S. Some will take this disparity as evidence that the French are better citizens of France than Floridians are of the U.S., they participate more in their civic process.
But, the lecturer continues, what if voting is not all there is to it? In a way, voting is like Christmas. It's easy for any person to get all psyched-up for just one day, but aren't we supposed to keep the spirit all year long. That's just kind of tough, you know. However, if we go back to the national numbers, we find that Americans (overall) participate more in the process, in non-voting ways: volunteerism (public area upkeep, local charity), activism, and vocalism. The French and much of the rest of Europeans, despite their very high voter turnout, do not take it much further (again, overall), content to make their choice at the polls, and entrust the remainder to their elected officials. In my opinion, this has a lot to do with more socialized systems in Europe. I prefer the American way, but then I am an American, so where's the surprise?
Some things that are, or have been, considered civic duty: