A Core War is a way of the Virtual Adepts to settle their conflicts. Similar to a duel, first, the challenge has to be issued and accepted by both participants.
A Core War is fought within the Digital Web. It's complicated, dangerous, involves complicated and dangerous viruses, and should always be announced in advance, because many who stumbled into the midst of such a war are fortunate to escape. The actual combat manifests itself according to the rules and the Adepts involved. Recorded examples include giant spaceship combat, armies of skeletons, and most or all of the apocalyptic imagery you can think of.