Ents and Huorns. There are plenty of talking trees in mythology but nothing like the Ents.
Tom Bombadil is original to Tolkien. Although I would argue that he is a nature spirit, similar to some of the Jotun, of Scandinavian lore. He appears to be a benevolent trickster, who is a manifestation nature itself. His wife Goldenberry is said to be a river spirit.
Hobbits are also original to Tolkien unless you class them as fairies or any kind of "little people" like leprechauns, or the hill people of ancient Britain. But Hobbits seem to be completely unmagical, simply a race of small humanoids who live in holes in the ground and live long lives.
Orcs and Uruk-hai are original to Tolkien and appear to be based on goblins and demons. Wikipedia says "Orc is from Old English orcneas, which appears in the epic poem Beowulf, and refers to one of the races who are called the offspring of Cain during the initial description of Grendel".
Fell beasts the Nazgûl Steeds seem pretty unique. They seem to be sort of horse-like with bat wings. Sort of like evil pegasi.
Elves, dwarves, trolls, goblins, wizards (magical practitioners), and giants are all pretty common in Scandinavian and Germanic mythology.
The name Balrog was original to Tolkien but they seem to be a kind of demon.
Barrow-wights, wraiths, and various kinds of fighting dead (including fighting skeletons), and vengeful ghosts exist in Norse mythology.
Most of the rest just seem to be large talking animals, which are pretty common in mythology and folklore everywhere.