The concept of cores could probably be considered a new feature in Omega. Instead, really, it's just the concept of weapon specials wrapped inside new packaging.
Remember EpicDuel Pre-Omega? So some weapons had special effects- let's take the Azrael's Bane. It had the Mark of Azrael, which meant that each hit with that sword had a 5% higher chance to achieve a critical hit. The concept of cores is that you can take that weapon special and slap it on your other (in this case, primary) weapons. Oh, and armors can have cores, too. Cores basically are that little trick up your sleeve (that you paid 7k Credits/495 Varium for).
So, how to put a core on your weapon? Well, select the weapon first.
Underneath the stat points and damage, you can see two apparently empty dotted boxes. One has a lightning bolt icon and the other, a skull icon. The lightning box represents passive cores and the skull box represents active cores. Passive cores are always effective during battle; they're like passive skills (hello Blood Lust). Active cores, on the other hand, can be used once and only once per battle. You can do that clicky trick with them.
Anyhow, click an empty box:
PS: I don't know how to remove cores.