Note that Middle-Earth Elves' abilities appeared magical to men but strictly there was nothing truly magical about any elf. Their ability to appear ageless to mortals was as a result of their "natural immortality" - to be immune to diseases, recover rapidly from wounds, heal others, "sleep" as they move, and as a result live a lot longer than all bipeds. They could however be killed like any men, and when dead would return to the earth (Valinor). They also possessed legendary abilities with bow and arrow, remarkable agility, heightened senses, and affinity with all of nature. They represented areas of perfection in a race that Man could never match.

Two answers here. First, innate magical abilities, has been answered previously. Elves are remarkable when compared to humans (or indeed any other similar race in Tolkien's mythology) but are not in themselves magical - more like mystical. I am however, reminded of the quote by Arthur C Clark (I think) which states words to the effect that any technology which we do not or cannot comprehend may be believed to be magic. Elven abilities, as already mentioned in the previous answer, could be thought to be 'magical', but are inherantly of them, and come with being elven. Second, items of magic. Here there is no doubt that elves (and especially the Noldor) had magical ability (although perhaps they would not call it that - see the chapter 'The Mirror of Galadriel' in FOTR). Galadriel's mirror is one of the few examples given in LOTR, but there are others, most notably the Three Rings, Vilya, Nenya and Narya. The three gave the wielders power to heal, support and build, (or enhanced their inherant abilities and knowledge). The Silmarillion is simply stuffed full of 'magical' elven creations. Feanor, the greatest of the Noldor (and uncle to Galadriel) created the three Silmarills, over which the 500-plus year conflict that drowned Beleriand was fought, and also (perhaps) the Palantiri. During the First Age in Beleriand a Sindarin elf named Eol made the sword Anglachel, later renamed Gurthang, which was sentient, and actually spoke at least once to it's last wielder, Turin. During the Second Age, Celebrimbor, nephew of Feanor, created the Three Rings, and, with the help of Sauron (disguised as 'Annatar') forged the nine rings for men and the seven for dwarves. Once the Third age was underway, and with the continuous leaving of those powerful (usually Noldorin) elves that remained, it is perhaps understandable that fewer items of this kind of power are mentioned. The only example of magic not bound in an item I can think of is the protection Galadriel afforded Lorien - 'I say to you, Frodo, that even as I speak to you, I perceive the Dark Lord and know his mind, or all of his mind that concerns the Elves. And he gropes ever to see me and my thought. But still the door is closed!' This is an echo of a previous magic from the First Age known as the Girdle of Melian, which prevented the entry of undesirables to the forest kingdom of Doriath. Galadriel lived for many years in Doriath (she met and married Celeborn there) and learned much from Melian before the destruction of the Kingdom by the Dwarves of Nogrod.

Elves have the ability to eat cake at rapid speeds. Often times elves dance in the moonlight while listening to a fat rodent play the keyboard, they especially love to dance after eating cornbread and wooden spoons.