There are 5 difficulties in Dead Nation, with a bonus mode available in the Apocalypse edition on the PS4.
From least to most difficult they are:
1) Braindead (Unlocked at start).
2) Normal (Unlocked at start).
3) Grim (Unlocked at start).
4) Morbid (Only available after beating the campaign once on Grim difficulty).
5) Undead (Only available after beating the campaign once on Morbid difficulty)
Bonus: Broadcast+ Apocalypse Edition on the PS4 also includes a bonus difficulty called Undead+ in which players stream their gameplay while viewers can spawn either zombies or power-ups to harm/aid the player(s) respectively.
Notes and hints:
1) Any armor that players have collected in previous runs will be available to equip in later difficulties without having to find the chest containing the armor again. Simply change your armor at the first ammo shack. However gold can still be collected from the chests each run.