Player Versus Player is often referred to as PvP.
There are 2 forms of PvP, ranked and unranked. In unranked you do not lose or gain anything, it is just for fun. In ranked you gain SS depending on a variety of variables including how many units you kill. The winner also receives a PvP trophy, used for crafting charms. The amount of workers you have in your medical building determines how many units you can revive after a battle. If you lose more units then you can revive, the extra units will "die" and the workers assigned to those units will need to be converted again. You can only do ranked PvP if you are in an active convergence. Owning territories in your convergence grants you a patria bonus and lets your gatherers collect ingredients for you.