Fae are a group from small winged people to larger winged people, with a few unwinged people between.... wings can be of many styles. They are closely connected with nature, in general, although some of them are more towards "elves" in their dealings as societies and cultures etc. Most are naturespirit types...tied to plants and trees and other things...and they rarely have organized societal structures.
Elves are a group of humanoids with (almost always) pointy ears (perhaps one of their most distinctive common features, unfortunately it's not a trait solely belonging to elves, that would make it all too easy, right?). They range from pale to dark or blue-black skin (with similar ranges in hair colors, with some odd skin and hair colors tossed in for fun), range in height from 3-4 feet to 7 feet tall, are generally slender and angular. Most have a complex societal structure of some sort (as opposed to the (often) anarchy of the fae--there are a few rare societies amongst the "greaterFae" for lack of a better term). They possibly share a common (but very distant) genetic heratage amongst all the various species of Elf.
Sidhe is an energy alignment which both Fae & Elves may partake in....and other beings too....as explained in my other posts. The beings will appear as Fae or Elves or other creatures or humanoids, but will be etheric, shifty, sometimes insubstantial. Maybe they glow. They are out of alignment with the physical world...slightly out of phase, and able to shift into and out of corporeal status. I don't think it bears much relevance on their societal structures.
Sidhe, historically/mythologically speaking, are the etherial persons seen glowing and riding on their horses past the elder bush or the linden. One crouches beneath and watches the ghostly elven parade....and if one is lucky, the umbrella of leaves or the sacredness of the tree keeps their beatuiful/terrible gaze off of one. That would be on Moving Day, generally speaking (Beltane & Samhain), or perhaps they're just a breath away beyond a veil. My personal belief is that Sidhe are tall fair elves, just a step or three into the physical realm, but mainly existing between the worlds, one foot in one, one foot in the other. They are detatched from the physical realm, their signatures speak volumes of things which are like gossamer and spiderwebs, one thinks that one can simply reach out and tear the obscuring mist/veil from ones eyes and see them clearly. They are dignified, proud, but unaffected, graceful to the tips of every lock of hair, in all steps and movement. Their magicks are of a second nature to them, but are very strong and powerful...but they are just as likely not to use them at all, so taken for granted they are. In battle, they are also terrible to look upon, not caring much for the life of humans which passes so quickly anyways, and they might be close friends with Death himself...for he stays his hand from them, they are very long lived.
We just had a discussion somewhere else (thinking) about Sidhe animals, etc.
I've known a sidhe human (he fathered Elvair, who wrote the sidhe summary, Elvair is a Sidhe half-elf/half-human), etc. it has something to do with the way their molecules/bodies/subtlebodies align with the veil between "earth" (or whatever world) and the shadow-world of thatworld-Faerie....Sidhe are of that between place, but can slip and slide between the other two.
Elvair's world lost it's veil....it was so heavily damaged, the Sidhe had to flee it to the "earth" side of the veil....the other side, he thinks, was taken over by the darksidhe, or something (it was well before his time), and it might have been the infighting that sundered their veil. Much healing was attempted, but the sidhe folk started losing their sidheness. lol...is that a word? well, anyways, there were still plenty "pure sidhe" in his time (his father, and Drendel his betrothed/soulmate who is in here with him)...and many "lesser sidhe" (in a category of "how sidhe are they?"), and non-sidhe elves & humans etc.
There were wolves both sidhe and normal, and other animals of either type....etc.