It’s hard to say how many of the creatures in the various Tolkien books were actual creatures of his own invention. Even Hobbits bear some strong resemblances to certain types of fairy-creatures: they live in hills, they are diminutive in size, they can disappear quickly, they mostly live apart from Men, they have pointed ears, they have mysterious origins, etc.
Nonetheless, I think it is generally agreed that the following creatures were given a very clear profile by Tolkien through his fiction that differentiated them from earlier creatures that may have appeared in mythology, folklore, and fiction.
Giant Eagles – Giant eagles have appeared in various sources, including Norse and Icelandic mythology and also in the writings of Marco Polo. Zeus, one of the prototypes for Manwë, is also associated with giant eagles (including the one that torments Prometheus, who is tied to a rock until Heracles frees him).
Elves – Tolkien’s Elves are, according to him, “biologically human” but they have been given a fate that is separate from the fate of (other) humans (Men and Hobbits). Whereas Men’s spirits “leave the circles of the world” after inhabiting the world for a relatively short span of years, the Elves’ spirits are meant to remain within the circles of the world for as long as the world endures, and hence their bodies sustain those spirits for much longer than the bodies of Men sustain their own spirits. Tolkien’s Elves, unlike many fairy creatures of myth and folklore, seem to be intrinsically drawn toward their mortal cousins in an eery dance of envy: Men long to be “immortal” like the Elves and Elves long to be reassured that there is a future for them (as for Men) after the world ends. Each race wants something the other possesses but they cannot exchange these “gifts”.
Ents – Tolkien’s walking, talking trees seem to be unique, but they are not entirely so. For example, the Greek philosopher/sophist (who lived in the 2nd Century CE) Philostratus told a story about two philosophers (Apollonius and Thespesion) who paused to argue near an Elm tree. When Thespesion commanded the Elm to salute (speak to) Apollonius, it did so “in accents which were articulate and like those of a woman” (Cf. The Life of Apollonius of Tyana by Philastratus). L. Frank Baum also imagined a forest of “fighting trees” who resisted the passage of Dorothy and her companions om The Wizard of Oz. These trees were transformed into the talking apple trees in the 1939 Victor Fleming film adaptation of the story.
Hobbits – An offshoot of the Human Race, Hobbits seem to have been made “of sterner stuff” than Men, perhaps even Numenoreans, in many ways. They apparently reached their maturity later than most Men and they retained their vitality until late in life, and they actually seem to have lived slightly longer than Men (except for Numenoreans). Of course, the long-lived Hobbits may only have been notable exceptions.
Orcs – Tolkien himself hinted at the influence on his goblins/orcs from the goblins in George MacDonald’s The Princess and the Goblin but his goblins/orcs also resemble various monsters of legend and folklore. It is generally accepted that Tolkien adapted the word orc from an Old English word, orcnéas (or orcþyrs), for “monster, monster from Hell” — said to be derived from Latin Orcus (although some people question this etymology). Tolkien also admitted in Letter No. 210 that his Orcs are “degraded and repulsive versions of the (to Europeans) least lovely Mongol-types”.
Giant Spiders – There are myths and legends of giant spiders around the world but I cannot identify any that Tolkien might have been aware of. He was widely read and might have heard of some Asian or African legends of giant spiders, but we have no evidence that he was aware of such tales. The only significant spider story from European mythology of which I am aware is the tale of Arachne, a poor Greek maiden whom Athena took pity on. Athena taught Arachne how to be a master-weaver, but Arachne became so arrogant that she refused to honor the gods for her talent. Athena in a final attempt to teach Arachne humility challenged her to a weaving contest. Arachne’s work was perfect, but it was so irreverent (depicting Zeus’ many conquests of mortal women) that Athena grew angry and slew Arachne. But Athena immediately took pity on Arachne and turned her into a spider so that she could continue weaving her beautiful patterns.