BEGINNINGS OF FANTASY
The source of fantasy art surely must start with defining fantasy. In the d ictionary it is defined as 'fiction marked by highly fanciful elements.' In my opinion it is anything that contains elements that can not be real in any shape or form. A unicorn for instance. There is no such thing, nor, to our knowledge, has there ever been a horse with one horn on its head (though it must be pointed out that there are other animals with one spiraling horn in the center of their forehead).
But where did it begin?
I will not go so far back as to cave man, but to the origin of modern fantasy which is very likely Greek, Roman, and other European, as well as oriental, myths and legends. All of these were created to explain things in nature and life. Apollo pulled the sun across the heavens. Zeus threw lighting bolts at the earth. Persephone spent six months in Hades every year. They explained the suns daily journey, the lighting during storms, and the seasons because the ancients did not have the ability to explain it until later scientists figured it out.
Their myths also had symbolism's. Phoenixes where rebirth. Griffins strength. Nymphs innocence and joy. Fairy folk mischief and mayhem. They even had those that told the tales of darkness, the fears embodied in vampires, ghosts and demons.
CREATURES IN MYTH AND LEGEND
Again, look at unicorns. There were several forms of the unicorn. One, the ka'lin, was a Chinese unicorn that more closely resembled a goat and brought marvelous luck to those who it chose to reveal itself to. Another, from Europe, is a creature with a horn; base white, middle black, tip red. The head was red and it had the hind legs of an antelope and tail of a lion. It was the symbol of purity. Only a maiden could approach it and its horn was sot for its restorative powers, something we would now call magic.
Gnomes, fairies, elves and other such things had their origins in England, Scotland, Germany, Finland and the surrounding area. Elves where only a few inches tall, and trolls where not the gruesome things we have now that turn to stone in the sunlight. People have lead off from the original tales and created them into something
more.
Mythology itself is a great influence on both art and writing. Take for instance the nymph. In Greek myth the nymph is a demi goddess of the trees, rivers, mountains and other such things in nature. They were the daughters of Poseidon and the handmaidens of Artemis. They protected the trees and in a way were the souls of the trees.
Now look at art. Many statues and paintings have been done of Apollo and Daphne, the girl who was turned into a tree. Many sculptures have taken trees and found the natural 'woman' within it and given it life. Others have merged the two so that one becomes the other. Painters take pleasure in forming a tree or fire and giving it a 'spirit' by using the paint to form a woman within the patterns of the object they are painting.
Dragons are quite a subject in Chinese art. A harbinger of luck and good will. A whole year is devoted to the dragon. The Chinese dragon is completely different from dragons you will find in other areas however.
The best known dragon is that of 'St. George and the Dragon' of course. I found nine depiction's of him on my clipart CD alone. How many more are there?
Some would say that the dragon of St. George was created as an allegory of Satan. Others that it was a large lizard that lived back then and was hunted to extinction. Truly there are those who believe that the dragon was a type of dinosaur that lived through the ice age and the flood. Personally I think it looks a little like a kamono dragon.
MODERN FANTASY IN LITERATURE
Modern Fantasy includes all those TSR stories and pictures, D&D especially. It includes the dragons of modern tales including Pern and Acorna and other such lands. Piers Anthony has given us Xanth and Maggie Furry gave Arriana. All of these are filled with magic, wizards, and creatures both from myth and from someone's creation. They blend the old and the new in ways that entertain and delight their readers.
Modern fantasy also includes J.R. Tolkien though I would not name him the father of fantasy. I know this is a general view of him and I will pull some flack for saying it but I found his writing very dry. Yes, he made elves a race of people and created hobbits but other then that what did he do? He stole elves and gnomes from mythology and gave the dragon a personality.
BUT! There is one story that I know of, much older then Tolkien and, in my opinion, is closer to starting fantasy. It is a legend called 'Beowolf'. For those of you who do not know the story it is of a man named Beowolf that kills a monster named Grendel and a fire drake that was toasting a town... isn't that something like the dragon toasting a town in 'The Hobbit'? Only the drake didn't have a personality. If you have never read this before I suggest Beowolf in modern english and Beowolf in the old english epic version. Both have the whole story and the fist is illustrated. Beowolf is a long tale but well worth the read.

Beowolf was first told by mouth and finally written down. I do not think the original author is known, nor the one who first wrote it down. As most stories and legends from the past it lived only because it was so fascinating to those who heard it that they continued to pass it on until someone finally decided to write it out.

Another, even older story that was actually written down, is told by Homer. 'The Illiad' and 'The Odysseyessy' are both epic poems as Beowolf once was. Both are based loosely on actual events, there was a city of Troy and there was a war between Greece and Troy. I believe there may have been a Trojan horse, so it is safe to assume most of 'The Illiad' is based (however loosely) on actual fact. However, 'The Odysseyessy' is complete fabrication. There is in it sirens, goddesses, transformation from man to pig and back, sleep spells, Cyclops's and other such things. In archeology they are tracing the path they think Odysseyysseus may have traveled and finding that it is possible that there was an actual voyage, but Homer exaggerated and livened it so much that it became a work of fiction.
LITERATURE AND MYTH TO ART
Shown here are two paintings from mythology. Medusa as seen on Perseus' sheild, and Hector killing Achilles. Both are age old stories, one told in 'The Illiad'. Painters and sculptors both have taken myths and legends and incorporated them into their art.
'The Victory' statue is of the goddess Nike, made to commemorate a victorious war. The Parthanon was a temple erected to the goddess Athena and even the pediments where sculpted into scenes from the tales of her life. 
Cherubs are a common subject. To the right is a satyr, as far as I know they are not very common though I have found one of a satyr and a nymph embracing.
Each of these are depiction's of the myths of lore. Even today you can travel through elfwood and find those who have depicted Perseus, Achilles, or Apollo. Greek and Roman mythology has influenced art from the beginning.
MODERN ART AND LITERATURE
Though mythologies influences can surely be seen in today's literature and art work we can also see that the changes have been quite dramatic. Unicorns are no longer so colorful, now usually a plain white horse with a spiral horn. Mermaids of all types can be found including those with legs and finned feet, those with tails, and those who can change from one to the other. Dragons very from the wise elders of Chinese lore to the crazed, greedy beasts of English tales.
Elves can be found to be two inches tall or six feet tall depending on the artist behind it. 
For the most part artist have taken their poetic license and run with it. It has resulted in such a wide verity of creatures that to track them all would take volumes. Even the human in fantasy has developed into clans, races, tribes, cultures and legends all their own. They have transformed, evolved, and metamorphosed in any number of ways.
Though their roots may show at times, it is quite safe to say that originality is sought after more often then not, and poetic license may often turn the cute two inch elf of mischief into the ranger type fighter or mage of newer tales.
BOOKS
To be completely honest I used no books in writing this. All but one picture on this came from a clip art CD, Printmaster Gold. Most of this I wrote from memory having read 'Beowolf' and 'The Illiad' when I was younger as well as 'Ulysses' which is the modern version of 'The Odyssey.'
I recommend any and all mythology. I only covered Greek myth with a bit of the orient but I neglected Egypt, Norse, and several others. I also suggest art history books that show the progress of art through the ages. The only fault I have found with these books is that they disregard modern fantasy art for the most part, but much can be learned from them anyway.