PVE: Nearly nothing. You will restart in a safe house. If on a Story Mission, you will restart at the last checkpoint. The npcs are also reset so you have to face the same opponents again.
Non Rogue: You will lose all unsaved Dark Zone loot items, some Dark Zone credits (~1700), a few rank Points (~1500-2000) and keys (normally i use just a single key, even if i carry ~40).
Killing 10-15 mobs will get you to the same DZ credit/rank level again.
Rogue: Same as non-rogue, but a higher amount of rank and dark zone credits.
After Patch 1.02: The lost amount of rank and DZ credits was dramatically decreased, it doesnt hurt too much, even if on manhunt now.
If you are rogue, your rogue timer will pause if you are attacking other players. That also applies to the 5 min counter when in Man hunt mode.
Losing Dark Zone ranking can mean a dark zone level downgrade!