As I show below, typical alien depictions include an elongated, pale face, two dark elliptical eyes, and a small mouth. Who first used this design for an alien? Where does this design come from?
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Arguably it all started with the Outer Limits episode "The Berrello Shield" from october 1964.
According to wikipedia, the grey aliens, also known as "Zeta Reticulans" rose to fame following the alleged abduction of Barney and Betty Hill in New Hampshire in 1961. The couple underwent hypnosis in 1964 to retrieve memories of the abduction and it was from these sessions that the following sketches were produced:
The topmost sketch is Barney's own, while the lower two were performed by David Baker based on Barney's description. In 1990, abduction-debunker Martin Kottmeyer remarked on the resemblance between the alien described in the 1964 hypnosis session and the alien in the Outer Limits episode that had aired two weeks before the session:
The Barney and Betty Hill abduction received great publicity and is acknowledged as the start of the grey alien trope, but it was probably this character from a TV show that was the original inspiration.
We can see how artists, from the very first second-hand sketch, have made free associations from the descriptions, bringing in their own visual memories. The David Baker sketches for instance, lack the completely black eyes, thin body and oversized heads that later foetus-inspired artists would use. Note that not even the abductee himself drew eyes like that! The artist David Baker instead seems to lean toward the face of a burn-victim.
Lasse Gustavsson, former firefighter and present motivational speaker, is probably fine with illustrating "face of burn-victim". Note how the nose resembles just two slits, like in the modern grey alien.
While @Abulafia's is wonderful, the Outer Limits was not the origin of the archetypal "grey alien". The Outer Limits simply helped to proliferate a pre-existing image.
The first description of such extraterrestrials appeared three decades earlier in Swedish author Gustav Sandgren's novel Den okända faran: En framtidssyn (The Unknown Danger: A Vision of the Future). It was written under Sandgren's pen name "Gabriel Linde".
In Sandgren's novel, we find the following familiar description of a race of extraterrestrials:
"...the creatures did not resemble any race of humans. They were short, shorter than the average Japanese, and their heads were big and bald, with strong, square foreheads, and very small noses and mouths, and weak chins. What was most extraordinary about them were the eyes — large, dark, gleaming, with a sharp gaze. They wore clothes made of soft grey fabric, and their limbs seemed to be similar to those of humans."
(Source: "Grey Alien")
The novel was accompanied by illustrations of the aliens. Unfortunately, I can't track down the illustrations, but Sandgren's written description is clearly the prototype for the modern "grey alien".
Authors and illustrators — Sandgren included — may have been influenced by an even earlier (non-fiction) article by H.G. Wells. In 1893, Wells wrote "Man of the Year Million", in which he projected that humanity would evolve into a race of short, grey-skinned beings with enlarged heads and eyes.
Wells reinforced this with the Morlocks in The Time Machine (1895), through which this image became synonymous with "highly evolved". However, since it was Sandgren who made the leap of applying this to extraterrestrials, I contend that the credit for introducing the "grey alien" should lie with Sandgren.
It's hard to pin down. Wikipedia's article on Grey Aliens states that
The origin of the idea of the Grey is commonly associated with the Barney and Betty Hill abduction claim which took place in 1961, although skeptics see precursors in science fiction and earlier paranormal claims. The Grey aliens are also famous from earlier depictions of the Roswell UFO incident from 1947.
In other answers and comments on this page you can find a number of possible earlier references, including this from keshlam:
It's worth noting that the behavior of the aliens as described by abductees pretty closely follows the behavior of the Fey Folk as described by traditional tales, with a think veneer of science slapped on it. Which I'm sure convinces the believers that that the elves were aliens, but which convinces most of us that aliens are just the modern version of blaming everything on the unseelie.
Given how far back these kind of sightings go, and how common the grey alien is among current UFO sightings, it's at least possible that this appearance is not something that some sci-fi author came up with, but instead the actual appearance of actual aliens. Or fey-folk. Or extra-dimensional visitors. Or future humans come back in a time machine to fulfill a cylical paradox requiring their presence. Or something.