Warframe

Nightmare Mode is a set of random challenges that increase a mission's difficulty.
Mission nodes under the effects of Nightmare Mode are marked with a swirl symbol.
The missions are selected randomly and change every day. Not every planet will have a nightmare mission on every single day.
Having completed several planets will thus speed up the completion of this achievement.
The game randomly selects missions for Nightmare Mode from any planet (except Mercury), provided you have completed all the missions on the given planet.

Just unlocking all the missions is not enough, you have to actually complete them all, this means you have to complete the Archwing missions as well.

To unlock this achievement you can replay the same mission 10 times.

Possible effects of Nightmare mode are:

No shields - self explanatory
Timer - missions are on a 5 minute timer, 1 kill will add 5 seconds to the timer
Health Vampire - you will lose health every second, killing an enemy will restore part of your health
Energy Drain - you will lose energy every second
Lethal explosive barrels - Exposive barrels will deal a lot more damage
Longer Reload - reload time is increased by 50%

In addition to these effects there is also a 50% chance that of low gravity taking effect. This will effect your movement and also the enemies.

The enemies on these missions will be high level, i recommend you only take lvl 30 Warframes on these missions and lvl 30 weapons as well.
Because of their powers and abilities Warframes like Rhino, Loki, Valkyr and Saryn makes these missions a bit easier.

I do not advise playing these missions on your own, teamwork makes them a lot easier to complete.

Nightmare mods are a possible reward for completing these missions.
These mods have dual effects (for instance higher fire rate combined with higher critical chance)

*edit*
credit to umpa lumpa 1234 for this:

Nightmare alerts might show up as well, these to do not require you to complete all mission nodes on a planet.
All the other Nightmare rules do however apply.
Waiting for these alerts to pop up could take a long time since they're completely random.
Here you can check past and present Nightmare alerts: http://goo.gl/Tmif7z

Nyari's guide is great for unlocking and understanding nightmare mode, but I wanted to add some tips for completing nightmare missions.

--You should have a passing understanding of the Damage 2.0 system. The full up-to-date system is here: http://warframe.wikia.com/wiki/Damage_2.0 .The system is very complicated, but a basic explanation is:
Yellow (armor) health bar: weak to Puncture, toxin, and corrosive(Electric+toxin).
Blue (shield) health bar: weak to Impact, electric, and very weak to magnetic(electric+cold).
Red health bar: Weak to Slash, Fire, and blast(Fire+cold).

My general loadout is a primary weapon with puncture and corrosive, a secondary with magnetic and maybe impact, and a melee weapon with Slash and Blast. This will give you some ability to cover all situations. However, the nature of this set is that you will need to control the range of fights to match the weakness of your enemies. Having to run up to every red health bar enemy could be deadly on death detonation.

--Overshields still apply in "no shield mode." The easiest source of repeatable overshields is The Taxon sentinel's Molecular Conversion effect. Shield restore items will instead provide overshields, but are less effective than health restores. A couple warframes have abilities that create overshields, but those are also the warframes with the least health and armor.

--the rhino warframe is easily acquired and makes no shield mode a joke. The parts can be acquired from the assignation mission on Venus against the Jackal. Repeat until you have all three parts (chassis, neuroptics, and systems) and the blue purchased in the market for 35,000 credits. Rhino's iron skin ability absorbs damage like a shield. With no mods, at level thirty, it absorbs 1200 damage for 50 energy and lasts until depleted. Rhino's armor value x 2.5 is also added to the shield. Since armor prevents health damage, you'd want it high for no shield mode anyway. Steel fiber is a common mod to increase armor.

Rhino is also solid on "Energy drain mode." if you equip an energy efficiency mode, you can cast Iron Skin at the start of a mission or when you find an energy orb in a container. Rhino doesn't need his other abilities and tanks ok without Iron Skin, so he is less affected by energy drain. Rhino can ignore "death detonation mode," too. With Iron Skin on he's immune to status effects. Iron Skin will deplete fast, so keep your distance.

--Valkyr has almost no shields normally and very high armor, so she is less affected by no shield mode. However, her basic frame is harder to acquire and she has a playstyle that requires practice. As I'm writing this, however, her Prime (improved) version is one of the easiest to acquire, so you may want to try her.

Inaros and Nidus both start with no shields, and therefore, suffer no penalty on the most common nightmare missions. Inaros is acquired from the Sands of Inaros questline, which is acquired from the biweekly trader, Baro Ki'Teer.

--Some Warframes that you won't acquire until very late in the game have abilities that can be useful in nightmare mode. Equinox, Harrow, and Oberon have some healing abilities.

--For nightmare missions where you just want to run by the enemies, such as death detonation, Nyx or Loki are excellent. Nyx's parts are acquired by killing Phorid during infested invasions. Her Mind Control and Chaos powers force enemies to fight themselves. Loki's Decoy or invisibility powers also let him slip by. These are harder to use than Rhino, but have more use in the outer planets.