Falkor (Fuchur in German) is a luckdragon, and a friend of Atreyu. He is the only luckdragon to appear, although five others are mentioned in passing. He helps Atreyu in his quest to find a cure for the Empress after escaping the web of Ygramul the Many.

Falkor has an elongated, wingless body with rudimentary paws and many white scales and hairs on the length of his body that appear pink in dull light. The form of his head is described less precisely, though his eyes are mentioned as being the colour of rubies. In the color illustrations of the original German version he appears much like an Oriental dragon. A cover for the book by Dan Craig illustrated Falkor as lion-like, while in the 1984 film adaptation of the novel, as well as its sequels, Falkor has distinctive canine features. Additionally, in both the first and second movies, Falkor asks to be scratched behind his ear, accenting his dog-like appearance, which led to confusion among those who have not read the book.

Unlike most of their dragon kin, luckdragons possess neither an immense physical strength, nor great magical talents, though they can still breathe fire, which is blue, as shown when Falkor fights Ygramul. Their only distinctive ability that sets them apart is their incredible luck in everything they do, as shown at one point in the novel when Falkor manages to locate and rescue his companion despite being lost in a violent, blinding storm and having absolutely no idea where to look for him. A luckdragon's typical answer to how it is going to accomplish a nearly impossible feat is always "With luck!" Despite such carelessness, it seems their luck truly never fails and is a valuable asset for their companions.

Another special trait of luckdragons is the ability to fly despite the fact that they have no wings. It is explained in the book that luckdragons are beings made from fire and air. When in flight a Luck Dragon is in constant motion and literally swims through the air. Luckdragons never stop taking in air and heat through their scales, which makes eating unnecessary and makes water deadly to them if they are immersed in it for more than a few minutes. Luckdragons are capable of sleeping while flying, and prefer to have as much open space as possible rather than be cooped up in a building, regardless of how spacious it may be.

Source – Wikipedia

The reason it became a Pedo-dragon is the phrase he uses when Atraya awakens after a long trip, stating I like children with a smile that can be defined as a Rape Face.

From here on there have been some video editing making it look like Falcor is really a pedo.
There are not yet many Meme's known for Falcor and there are some flying around the internet.