PVE stands for "Player vs Environment". It typically denotes any game play that involves players competing against the game itself (the "environment"). It is in contrast to "PVP" which is "player vs player", which involves game play where players compete against each other.
As per the description of Civony on its online forums by the game developers, Civony is considered a PvP game. There are some elements of the game that can be considered PvE, mainly due to the inclusion of barbarian towns.