Okay, in Scandinavian contexts, we don't have many depictions of álfar in manuscript illuminations, nor do we have many in textual remnants that are not Christianized quite a bit. Specifically, the Prose Edda of Snorri Sturlusson is our primary source for what we consider to be the 'classical' depiction of elves in Icelandic sources (and they appear almost exclusively in Icelandic sources.) Snorri's delineation of álfir into ljosálfar, dökkalfar, and svartálfar is in line with what we would now consider to be angels, demons, and dwarves, and may reflect a folk-belief in beneficial, oppositional, and neutral spirits.
That said, we have an extremely common alliterative element in Skaldic verse - Æsír ok Álfar - which seems to equate the elves with the Æsír, or gods or, at the very least, puts them on the same footing in terms of spiritual importance. Given this, it's possible - though by no means certain, since Old Norse texts are almost criminally light when it comes to the description of people, generally - that they looked similar to the Æsír, which is to say like regular people.
However, Tolkien was drawing from an Anglo-Saxon tradition and, while that's not my field of expertise, I do know a guy who's a great resource for it.
If you're up for some reading, Alaric Hall's PhD on Elves in Medieval England is available online, as is his article on Elves in Anglo-Saxon England. Both of them are fantastic places to get the information you're looking for and find some more sources (his bibliographies are goldmines)
For the actual discussion of "what constitutes an elf?" in folklore, mythology, and pre-contemporary literature, I'm going to bow to u/itsallfolklore - you might want to see their answers for Where did the lore of trolls, elves, dwarfs etc originate from? Are they from the same source or time period? and J.R.R. Tolkein single-handedly created our modern conception of staple fantasy races such as Elves, Dwarves, and Orcs. Are his versions based on earlier legends, or did he invent them wholesale? for starters, and I'm going to focus on some of the literary traditions of depicting elves leading up to Tolkien.
The Wild Things are somewhat human in appearance, only all brown of skin and barely two feet high. Their ears are pointed like the squirrel's, only far larger, and they leap to prodigious heights. They live all day under deep pools in the loneliest marshes, but at night they come up and dance. Each Wild Thing has over its head a marsh-light, which moves as the Wild Thing moves; they have no souls, and cannot die, and are of the kith of the Elf-folk.
At the time J. R. R. Tolkien was writing the background material for what would become The Hobbit, The Lord of the Rings, and The Silmarillion, the "elf" concept in English literature was largely synonymous with "fairy", and covered a wide range of supernatural beings, many diminutive. For example, in Shakespeare's A Midsummer Night's Dream, the elves are all implicitly tiny:
Come, now a roundel and a fairy song; Then, for the third part of a minute, hence; Some to kill cankers in the musk-rose buds, Some war with rere-mice for their leathern wings, To make my small elves coats, and some keep back The clamorous owl that nightly hoots and wonders At our quaint spirits. Sing me now asleep; Then to your offices and let me rest.
And this followed through into the 19th century with popular works like "The Elves and the Shoemaker" in Grimm's Fairy Tales:
As soon as it was midnight, there came in two little naked dwarfs; and they sat themselves upon the shoemaker’s bench, took up all the work that was cut out, and began to ply with their little fingers, stitching and rapping and tapping away at such a rate, that the shoemaker was all wonder, and could not take his eyes off them. And on they went, till the job was quite done, and the shoes stood ready for use upon the table. This was long before daybreak; and then they bustled away as quick as lightning.
In fairy tales as well as plays, there's a great variety in the depiction of elves - they tended to be diminutive, supernatural, and nature-oriented, but that was the extant of what you could say about them; it was practically a by-word for an entire class of minor spirits and entities as it was a specific designation, and not always well-distinguished form dwarfs. You can see the influence of this in fantasy fiction into the 20th century in forms like the popular depiction of Santa's elves, the Keebler elves, Lord Dunsany's "The Kith of the Elf-Folk," and Tolkien discusses them in his essay "On Fairy-Stories", where he notes:
The diminutive being, elf or fairy, is (I guess) in England largely a sophisticated product of literary fancy. [...] In any case it was largely a literary business in which William Shakespeare and Michael Drayton played a part.
But Tolkien's conception, drawing from further abroad and trying to marry the depictions of different myths with the contemporary English conception, yielded a more complicated elf. You can still see something of the "wild" appearance in an early depiction of Wood-Elves in The Hobbit:
The feasting people were Wood-elves, of course. These are not wicked folk. If they have a fault it is distrust of strangers. Though their magic was strong, even in those days they were wary. They differed from the High Elves of the West, and were more dangerous and less wise. For most of them (together with their scattered relations in the hills and mountains) were descended from the ancient tribes that never went to Faerie in the West. There the Light-elves and the Deep-elves and the Sea-elves went and lived for ages, and grew fairer and wiser and more learned, and invented their magic and their cunning craft in the making of beautiful and marvellous things, before some came back into the Wide World. In the Wide World the Wood-elves lingered in the twilight of our Sun and Moon, but loved best the stars; and they wandered in the great forests that grew tall in lands that are now lost.
Here, Tolkien is making a distinction between the nature-oriented, less stately elves of the forest and the more noble, civilized, powerful and polished elves which would appear in The Lord of the Rings, based on Nordic and Germanic depictions.
This wasn't a completely unprecedented effort on Tolkien's part, in 1924 Lord Dunsany had published a novel titled The King of Elfland's Daughter which similarly focused on more regal, magical, civilized elves in the Germanic/Nordic mold than the diminutive nature-spirits of "The Kith of the Elf-Folk," and Poul Anderson's novel The Broken Sword (1954 - one year before The Lord of the Rings) uses a very similar depiction, although his elves are necessarily very less scrupulous and somewhat degenerate in their tastes (not balking at incest or rape or stealing children, for example) - Anderson's novel went on to have a considerable influence on Michael Moorcock, who ended up writing the Elric of Melniboné series beginning in 1961 with the very cruel but highly civilized and sorcerous Melnibonéans as elves-by-any-other-name.
And while Tolkien, Dunsany, and Anderson's conception of the "noble" elf might be the most successful in fantasy fiction, and the diminutive elf of Shakespeare and fairy stories still popular in the form of Santa's helpers and the Keebler cookie elves, there was yet another trend worth looking at:
More particularly I became convinced that much of the folk-lore of the world is but an exaggerated account of events that really happened, and I was especially drawn to consider the stories of the fairies, the good folk of the Celtic races. Here, I thought I could detect the fringe of embroidery and exaggeration, the fantastic guise, the little people dressed in green and gold sporting in the flowers, and I thought I saw a distinct analogy between the name given to this race (supposed to be imaginary) and the description of their appearance and manners. Just as our remote ancestors called the dreaded beings 'fair' and 'good' precisely because they dreaded them, so they had dressed them up in charming forms, knowing the truth to be the very reverse. Literature, too, had gone early to work, and had lent a powerful hand in the transformation, so that the playful elves of Shakespere are already far removed from the true original, and the real horror is disguised in a form of prankish mischief. But in the older tales, the stories that used to make men cross themselves as they sat around the burning logs, we tread a different stage; I saw a widely opposed spirit in certain histories of children and of men and women who vanished strangely from the earth. They would be seen by a peasant in the fields walking towards some green and rounded hillock, and seen no more on earth; and there are stories of mothers who have left a child quietly sleeping, with the cottage door rudely barred with a piece of wood, and have returned, not to find the plump and rosy little Saxon, but a thin and wizened creature, with sallow skin and black, piercing eyes, the child of another race. Then, again, there were myths darker still; the dread of witch and wizard, the lurid evil of the Sabbath, and the hint of demons who mingled with the daughters of men. And just as we have turned the terrible 'fair folk' into a company of benignant, if freakish elves, so we have hidden from us the black foulness of the witch and her companions under a popular diablerie of old women and broomsticks, and a comic cat with tail on end.
Machen's darker conception of the elves is implicitly euhemenistic; it takes as its base the fairy stories of stolen babies and changelings left in their place, elf-shot, and the like, and tries to make a quasi-realistic depiction of them as flesh-and-blood entities - dim, strange survivals, primitive and alien and terrible. This wasn't a necessarily widespread literary tradition, although the idea was picked up by H. P. Lovecraft and Robert E. Howard, but it goes to show that "elves" weren't just limited to the stately figures of Tolkien or the diminutive sprites of the Brother's Grimm.
Good earlier thread on the subject: https://www.reddit.com/r/AskHistorians/comments/4f91zc/jrr_tolkein_singlehandedly_created_our_modern/