It is the bundle of obligations that come with belonging to a republic: voting, serving on juries, staying informed, showing up. A republic cannot run without it, and that is the whole point.
Civic duty refers to the responsibilities citizens owe to their community and country, the obligations that come with the rights of membership. Some are legal, like obeying the law, paying taxes, and serving on juries when called. Others are moral, like voting and staying informed.
It flows from the nature of a republic. A monarchy runs on the king; a republic runs on its citizens. If the people do not participate, do not vote, do not serve, do not pay attention, there is no one left to actually govern, and self-government quietly fails.
The founders were obsessed with it. Drawing on the classical idea of civic virtue, they believed a republic could survive only if its citizens placed the public good above private comfort, at least enough to keep the system alive. A republic of the apathetic was, to them, a republic doomed.
It is mostly unglamorous and unenforced. No one is jailed for not voting or not paying attention. Civic duty depends almost entirely on citizens choosing to honor it, which is exactly what makes it fragile, and what makes a society that takes it seriously so strong.