What is virtue and what is it use for?

Virtue medals are awarded at the end of the Clash of Provinces 4 weeks season based on your performance. You can use these in the inventory to turn them into virtue points.

You can then use the points in the Virtue shop to buy some in game items like speedups. The shop only appears on certain days though. You can find it (when it is open) by taping on the Military Intel building.