I have read that Tolkien was very influential in the history of certain mythical beings, such as the elves, but what did he take from previously established lore and what was of his own creation?
As /u/university_press suggests, Tolkien borrowed a great deal from literature - he was a medievalist after all - and from folklore. His forest elves in The Hobbit follow a well-known motif with their enchanted circle that disappears as soon as intruders arrive. And the abduction of those intruders is also straight out of British folklore.
His elves tend to follow the pattern of Welsh fairies, the Tylwyth Teg, who tend to be described as having the most regal of British faerie courts.
Tolkien deviates from the oral tradition with the way he has people interacting with the elves. Tolkien apparently liked the idea of the Fall of Adam, the idea that people "back then" were far nobler (and taller and longer-lived) than we could hope to be. He explores a world where the nobility of man (to use a sexist term in keeping with Tolkien's time) allowed people to interact more freely with elves. A by-product of this prohibits the strange, eerie nature of an encounter with faerie. Tolkien tries to capture this on occasion when his characters enter an enchanted realm, but after that initial moment of suspense, everyone meets one another and the enchantment is only a dim memory. "Hey, you remember Joe the elf, don't you?" "Oh sure, it's been a while, but I sure do remember him. Great to see you again, Joe." It's a bit too casual and commonplace.
Of course, Tolkien was a master author, and he pulls it off, but his interest in that antique setting, when the nobility of man rose to the level where people could interact with elves at will had its price, and the price he paid was the most profound aspect of "his own creation" as you ask. In this and in its ramifications, we see the clearest departure from folklore.
I'm sure someone else will come along and give a better answer from a Tolkein perspective, but having studied and read a bunch of Germanic and other literatures, almost everything that turns up in Tolkein was borrowed from elsewhere. Gandalf is a dwarf in Norse mythology, as are most of the actual dwarves (the rest sound like those Norse dwarves). The character of Gandalf is borrowed from the Finnish Väinämöinen. The Elvish language is based on Latin (Latin grammar and syntax). Even the kingdom names are based on historical objects; Gondor is very close to the Ethiopian kingdom of Gondar. The majority of Tolkein's influences were in his own field of study; his article on Beowulf is still influential, and included on many undergraduate reading lists. Even if any names or mythologies weren't directly taken from a pre-existing exemplar, they were heavily based on something real anyway.
Edit: A big part of what Tolkein was doing, was re-hashing old mythologies in a new way. The geography of Middle Earth is similar in many respects to that of Europe. This might seem odd to a reader of modern fantasy fiction, as the made-up element is more prominent, but Tolkein was engaged on a project of breathing new life into old myths, with his own creative element of course included.