Warning Spoilers Alert, Storyline Information from The Bureau
Beings of pure energy, without any physical substance. Homeworld unknown. Their appearance is of a tall, near-transparent 4-armed humanoid, with a dozen thin tentacles coming from its backside. The head is a large neckless dome that covers what is either a face, or a visible brain.
Two are encountered: a feminine-'voiced' electric-yellow one named Shamash (a Hebrew word that translates to "Sun"), and an electric-blue one named Asaru (possibly a Japanese word meaning "Look for"/Seeker/Seer), that speaks with Agent Carter's voice (though Shamash claims this is only because 'he' has not yet "know (his) own." She claims that they are beings of influence, and have merged with beings in multiple realities, with the intent of sharing information for the rare occasions that their kind meet. It is implied that they set themselves on planets that harbor the possibility of intelligent life, and simply wait for a suitable being to be born.
Shamash claims to be a much older Ethereal, who long ago merged with the Zudjari scientist who would become Origin. Eventually, through years of study, Origin found a way to not also break their bond, but effectively enslave Shamash, creating the Mosaic network through her psionic abilities, and enslaving his own people, leading to the Outsider empire.
The other, Asaru, has been sleeping on Earth for an unknown amount of time and, according to Shamash, was awaken before his time. He was in hibernation in an unseen artifact that was discovered in an Elerium mine in Bannack, Montana. The artifact was then in the briefcase that Carter had transported to Area 51 (Groom Range), before Carter is attacked by an Infiltrator/Sleepwalker, who attempts to capture the artifact. When the Infiltrator/Sleepwalker had near-fatally shot Carter and then began to strangle him, Asaru escapes and kills the alien and merges with Carter, and the player can now control Carter. Just outside, the invasion has begun.
Breaking the 4th wall
The implication is that the player is actually playing as Asaru (which explains the third-person aspect). When the Ethereal bond is eventually shown, it is by the tentacles being linked to Carter's body and joints, suggesting puppetry. And as Director Faulke says, no other Agent or Squad has the success rate that Carter has: in truth, it is Asaru supplementing the abilities and directing both Carter and his squad with the precision of a computer (ie: a PC, a PS3 or an XBox 360). This also opens the possibility that Carter's and the XCOM agents more unique abilities (Lift, etc.) are merely possible through harnessing the Ethereal's power through the Venn Brace.
Later, during the infiltration of the Outsider homeworld, Carter learns of the Ethereal's existence from Origin, and manages to capture Shamash with the Venn Brace (implying that the Brace and the artifact are of similar nature), crippling the "Moz Zaaik" network. And with the detonation of a "Nuculerium" bomb, Origin and the Mosaic core equipment (and much of the Outsider homeworld) are presumed to be destroyed. However, the years of study have given Origin enough knowledge to mimic an Ethereal's abilities, and continue to exist as one within Mosaic (though at a severely weakened state, at a fraction of a true Ethereal's power) in a backup core (that managed to escape through the Venn Gate that XCOM had used) on a Zudjari dreadnaught, holding the remains of the network up, and able to continue the invasion, even assaulting XCOM's base itself, Site X.
After Shamash awakens in XCOM's laboratory, Carter realizes that he had been controlled by Asaru (ie: you, the player), and destroys Shamash in a rage (while surprised that a handgun seemed to have worked in killing her), right at the start of the assault of Site X. Carter moves through the base while fighting Asaru's control, eventually threatening to set off a bomb to destroy them both, giving the player/Asaru 35 seconds to release him. After ejecting from Carter, Asaru makes the choice of merging with either Director Faulke, Doctor Weir, or Agent Weaver; the three humans, realizing that the Ethereals are telepathic, correctly assume that Asaru can 'hear' them, and give permission for him to merge with them.
With the new merger complete, Faulke, Weir, and Weaver escape Site X in the Avenger, and plan the assault on the faux-Ethereal Origin's mothership. The human that has merged with Asaru, having the ability of a full Ethereal, states that if they connect with the core of the ship (the backup Mosaic Core), they can overwhelm and delete Origin for good.
When the Mosaic core is infiltrated (and while an infinite stream of regular and Elite Mutons swarm in), the linked human will use Asaru's power and give the order for all Outsiders to stand down, ending the invasion, and killing what's left of Origin. Depending on the human chosen, the Mosaic/Human/Ethereal will command the remaining Outsiders through one of three endings, revealed as the human talks to The Council of their actions, managing to keep the entire invasion secret, stating that anyone who knows the truth is either dead, or working for XCOM.
And when a Councilman asks of the "other creature," the human says that the Ethereal bond was broken shortly after final operations, and that "Its status, its location, and its intent are ... unknown." 
- It is possible that these Ethereals are the Ethereals ones who managed to ascend, while the ones that appear in Enemy Unknown are the aliens that failed to ascend and so were forced to live in a physical body.
- This doesn't invalidate the possibility that the they refers to a yet unseen species of aliens (who placed Asaru on Earth).
- The New One mentioned by the Uber Ethereal in Enemy Unknown could also refer to them expecting for Earth's Ethereal to reveal itself through the Volunteer and join them.
- Another possibility is that Asaru never left/returned to Earth during the events on Enemy Unknown and that someone in XCOM, possibly the Commander (the player), is merged with the Ethereal. This also explains the vision the Volunteer at the end of the game of Earth being destroyed as Asaru warning him/her of what is about to happen if the Temple Ship isn't destroyed.
- More credence is given to the theory that the XCOM Commander is bonded to an Ethereal in the final scenes of XCOM 2, where the Commander is shown to have extraordinary psionic abilities under pressure - controlling an alien Avatar in the final mission and then going toe to toe with three enemy Ethereals in psionic combat in the final cutscene. In addition, several brief glimpses of an Ethereal are seen as the Commander goes under to control the avatar.
Autopsy Report: Ethereal
WEIR’S NOTE: Due to unexpected interruptions, a full autopsy of the captured Ethereal could not be completed, and much of Dr. Heinrich’s notes have been lost in the ensuring chaos. Attached are what little has been recovered:
Telepathic abilities, limit unknown but able to subtly influence host. Possibly other psychic abilities such as telekinesis? Must be careful about interactions until we find a way to shield it.
Only partially exists in our plane of existence(?)- definitely must be more than the frail physical form apparent to our limited instruments.
Uncertain age- potentially ageless. Current specimen supposedly sustained in Origin host/network for millennia. If it can change hosts, an Ethereal may be able to pass undetected for generations on a planet.
Apparently very rare- only one recorded among all of Outsiders’ conquests, and apparently a target of great value. Potential applications for warfare unknown, but unpleasant.