Runes are permanently equippable to your character and replace the hack module system from the previous update. As you level, you will eventually be able to slot 3 at a time, which will allow you multiple options per match. You can only have 1 rune active at a time.
The last and final rune (Blood Trail) is unlocked upon achieving level 44 in Multiplayer. You can view these on the main MP screen or while in a lobby.
EDIT: Just a small tip, when you unlock the "Bounty" rune, make sure to play with that. It's unlocked at level 37, and it'll help you clean up those last levels pretty quickly.