Well, these are three adjectives for "something from the elves". But I'm Spanish and in my language there's only one adjective for these (élfico), and I can't understand what the difference is.
Okay, here goes: Etymonline's comments on elf reveal usage from the 1550s and stemming from the words elf, ælf, ylf. The plural mentioned there is "elves" which would match the traditional pluralization of
lf: shelves; wolves. Unfortunately, there is no word for wolven so that doesn't help us much.
The same site does have an entries for elven:
elven - O.E. ælfen; see elf) [sic] + -en (2). Apparently obsolete until revived by Tolkien.
elfin - 1590s, from elf; first found in Spenser, who may have been thinking of elven but the word also is a proper name in the Arthurian romances (Elphin).
elfish - c.1200, alvisc; see elf + -ish.
Tolkien's usage of elven strongly implies that this is the appropriate term for his elves. The c.1200 usage of elfish seems appropriate for mythological or physical references to elves (his ears are elfish; their attitude is elfish).
(Edit per Martha's comment): That being said, Elvish is also the Tolkien proper term for their group of languages with the prime examples being Quenya and Sindarin.
As far as non-Tolkien usage, Wowpedia consistently uses elven in their article on Elves. Dungeons and Dragons appear to use elven for the language and adjective. Other fictional universes could easily vary. It would be best to check with their experts.
The summary: Elvish for physical attributes or mythological references; elven for Tolkien and most other fictional universes. Elfic is unused in English and not advisable.
elfish/elvish use the common general adjective suffix -ish
I would recommend using either elfish or elvish as the corresponding adjective to elf/elves in general or if you’re uncertain. These words are formed with the common English adjectival suffix -ish which has a quite general meaning: they can mean “of an elf,” “like an elf,” “of elves.” So it’s hard to go wrong with one of them. (Using “f” or “v” doesn’t make a difference in meaning, only a slight difference in connotation: “elvish” is probably more commonly with “Lord of the Rings” or “Dungeons and Dragons” elves, while “elfish” may be more commonly used in contexts like “Santa’s elves.” Also, "elvish" may sound silly to some people due to the similarity to the name "Elvis.")
don't use elfic
As others have mentioned, “elfic" is basically never used in English. Using it would seem like an error caused by trying to translate the word directly from Spanish élfico.
elfin ususally has connotation of "otherworldly" or 'diminutive"
The adjective elfin is fairly common, but there are some nuances to its use. Applied to real people, it usually means “fine-featured,” probably with some kind of otherworldly appearance. In fantasy contexts, elfin can also be used as an adjective that is basically synonymous to elfish/elvish.
elven: originally a noun, later reinterpreted a an adjective
As other answers have mentioned, "elven" seems to have been popularized by Tolkien. The etymology and the way he used it make me a bit leery of using it as an adjective, but that’s just my opinion. It is commonly used as such in much of modern fantasy literature. Since it is associated with fantasy literature, it might seem a bit odd to use it in relation to "Santa's elves".
The Oxford English Dictionary entry for elven says it is a noun from Old English ælfen, ęlfen that was originally a femine form of the word “elf,” but later used to refer to any elf in a gender-neutral fashion. It says it is used in the works of J. R. R. Tolkien in the following ways: as a combining element that is appositive (the OED’s example: elven-kin) or attributive (the OED’s examples: elven-king, elven-tongue; elven-wise). So basically, “Tokien” used “elven-” as an archaic element for forming compound words with an appropriate feel for his fantasy.
There seems to have been some conflation of these compound words with sequences of adjective + noun, giving rise to the use of elven as an adjective.
Interestingly, the -en suffix of the re-analyzed adjective elven has also been applied now to at least one other word: modern fantasy uses the adjective dwarven, derived from dwarf/dwarves, as an alternative to dwarfish/dwarvish. I asked a question about the history of dwarven and it seems the first known use in a published text was in 1976, in the Greyhawk “Dungeons and Dragons” supplement; the form seems to have also arisen independently in some works of JRR Tolkien that were written earlier but not published until 1984 in The Book of Lost Tales, Volume II.
Don't use "elvic".
There is no big difference in meaning between elfish, elvish, elfin, elven. Any can be used for the general meaning "somehow related to elves."
Elfin is best if you're describing a real person and you want to convey that the person has a delicate, otherworldly look. E.g. if you're describing the face of someone who is not literally an elf, "elfin face" is better than "elfish face," and much better than "elvish face" or "elven face."
Elfish is probably not the best choice if you're talking about "Lord of the Rings" or "Dungeons and Dragons" style elves. E.g. to express the idea of "magic of the elves," it sounds better to say "elfin magic," "elvish magic," or "elven magic," and it doesn't sound as good to say "elfish magic."
Elven is probably not the best choice if you're talking about Santa's elves.
I don't think there is a word, elfic. You may be referring to the Old English (OE) ælfisc.
The words elfen, elfin, and elven are all spelling variations of the same word and when used as an adjective (or noun), can be swapped at will with no loss of meaning.
The root word is elf (from OE ælf), plural is elves. (Mark the f to v!)
After that, it gets twisted.
From OE, the original adjective was elfish (OE ælfisc).
In OE there was also elfen (plural was elfenne) ... a female elf, a nymph. That morphed in an adjective in Middle English with the original sense of feminine, child-like. It was also spelled elfin and elven.
During this time we also see the 'n' drop and a female elf is an elfe ... with the plural of elfen! Confused yet?
Naturally, if you have elfe, there is a spelling variation of elve (f to v)... with the plural of elven.
There are also compounds like elfsheen (adj. - supernaturally beautiful, noun - supernatural beauty).
So if you're keeping count:
There is only one male form: elf. (However, you can also call a female elf, an elf.)
There are four possible adjective forms: elfish, elfin, elfen, and elven.
There are five possible female forms: elfe, elve, elfin, elfen, and elven.
It's little wonder that folks are befuddled!
Elfin- Something from elves Elven- Elf like Note: The people who printed Tolkien's books changed elfin to elven and vice versa quite a bit, so there is no telling how accurate anything gotten from there is. I got this from the "note about the text" in my copy of LOTR, so if you want more info you could probably get some from the most recent version of that. Also, dictionary definitions would probably help you a lot. Also, I'm just using Tolkien because that seems to be the basis for a lot of these answers. Real definitions may not actually apply in books, so if you want clarification about actual meaning, a dictionary is definitely the way to go.