Stealth

With Update 7.0, new mechanics for stealth offer a better and easier time doing full stealth runs. This is only the first update of the like, so expect tweaks in the near future.

How to prompt an assassination
As long as the crew hasn't been alerted to your presence, you are eligible for performing stealth attacks. In order to do so, approach an enemy from behind (crouching for best effect), without alerting anyone to your presence, and press 'E' when the prompt appears. Depending on what Warframe and Weapon you are using, different animations will occur. Successfully pulling off a stealth attack will deal damage with a large multiplier (most likely different for weapon types, ie. short, long and heavy). The animation also varies with weapon type.

There are some bugs with the prompt. If you aren't at the same altitude as the target, the prompt wont appear.

Detection
Grineer are the most basic and least varied in design, and all Grineer can be assassinated. The following mostly refers to Grineer behaviour, and is an incomplete list.
 * Crouch for better sneak runs. You'll have a smaller model so you can hide your silhouette better. This is harder for larger frames like Rhino and Frost. The speed at which you move is also more convenient for timing your melee than running, and you will be less likely to bump into the enemy before the promt appears. Also, it seems that enemies are less likely to notice you while you are crouching.
 * Noise caused by your weapons or abilities will alert enemies (shooting, meleeing the environment, Decoy via the Loki warframe, etc.). Surprisingly, noise generated by breaking fans or storage containers or opening doors will not attract attention. Certain weapons make less noise (Paris; certain weapons with very good range like Latron and Snipetron, at the right distance), and thus are less likely to attract attention.
 * Their range of view is <180 degrees, so crouching still in a corner is a legitimate way to expose the back of a Grineer that just walked through a door. They are also bad at detecting objects at a distance and anything somewhat above them. If you move too quickly or stop crouching, and you might be detected. If you bump into them, they'll grunt and be alerted of your presence (bumping into somebody while invisible will alert). Hiding behind terrain will conceal your presense.
 * Performing the stealth attack and not killing the enemy will cause alarm (happens a lot when in a high level area, or on heavy Grineers with shields)
 * If you are ever forced to reveal yourself to something (camera, failed to kill, missed, etc.) you can still remain undetected if you kill/destroy everything before a console is touched, and before others in other rooms are alerted.
 * Cameras make loud alarms when they see you, but again, if they are destroyed and nothing is in range to take advantage of their alert, you're safe. Rooms with two cameras are very hard to get past without spending energy or alerting people.
 * Rule of thumb: kill it with one hit while nobody is looking in the general direction, and you're good. This includes cameras. If they see a corpse that hasn't disapeared yet, they will be alerted to your presence.

In the event of failure to kill something with a single stealth attack, or somebody is alerted, you can kill it and anything in the vicinity that knows of your presence. However, if this happens, the rest of the enemies on the level will be "on alert". They will hide behind doors and keep watch, without wandering. Enemies in that position are unable to be assassinated.

Holes in the mechanics
Since the concept of stealth is still a new one, there are some gripes that have not been addressed.
 * The practicality of attempting assassination on durable opponents (Shielded heavy Grineer, etc.)
 * Unable to attempt assassination on Moas, Ospreys, Ancients, Shield Lancers, Cameras, and Bosses. This makes the stealth approach a bad one against Corpus and Infected.
 * After each stealth attack, you finish standing up, which might reveal your position. You will need to release and re-press the crouch button to return to crouch position. If you are using a toggle crouch key, simply press it during the animation to return to crouching when the attack completes.
 * Sentinels don't cloak themselves so the invisibility they provide is less useful then it could be.
 * With the Paris, you can shoot an enemy with a one hit kill and no one around will be alerted, even if they are right next to them.
 * There appears to be no inherent benefits of stealth gameplay aside from the prompt for an assassination and lower enemy spawn rate, which is counter productive given that weapons level via kill exp.