"Core War is a game is which two (or more) programs compete for control of a virtual machine".
That's pretty much the standard descripition, but it's a bit dry: what does it actually mean?
Well, imagine a battlefield, empty apart from two warriors; these warriors aren't directly under your control, but follow instructions given by you before the battle started. Those instructions might be to look somewhere else on the battlefield to see if your opponent is there, to attack somewhere on the battlefield, or to make another copy of itself, so that if one copy is killed, the warrior as a whole will live on.
Obviously, the actualy rules of Core War are rather more formalised than that: programs are written in an assembly-like language called Redcode, and each warrior executes one instruction before control passes to the next warrior. Warriors can also divide their processes in half, so that they have two copies each running at half the speed (or three at one third speed, etc).
As with almost everything, the best way to learn about Core War is to try it! To this end, I have created a Guide to getting started in Core War.