The first discovery of dinosaur remains definitely occurred before the time that humans began recording history. The matter of what qualifies as being the first discovery of dinosaur remains is more a matter of terminology, such as when the fossilized remains that were being found were actually considered to be those of dinosaurs. The first humans to inhabit the earth undoubtedly found the remains of dinosaurs, and ancient cave renderings that have been found throughout the world verify this. These early accounts of humans possessing knowledge of the existence of the dinosaurs have caused many to speculate that some survivors of the mass extinction 65 million years ago actually shared the planet with early man, but there is no solid proof of this. Although the first person to find dinosaur remains did so thousands of years ago, it wasn't until the 1800's that scientists actually put a name on the creatures whose fossilized remains they were finding, the creatures were from then on referred to as the dinosaurs.
The first written accounts of dinosaur remains having been found date back to the time of the ancient Greeks, who tried to explain the fossilized remains by creating epic tales of giants and mythical creatures. Chinese writings by Chang Qu that described the discovery of dragon bones 2,000 years ago also point to the discovery of fossil remains. There is no doubt that these early findings mystified those who found them, and given the curiosity of the human mind these relics begged for an explanation. Literature and beliefs regarding the origins of life on earth were altered by the discovery of fossils before they were even identified as being that of dinosaurs. Mythology and superstition was built around these enormous and strange looking bones, and from those findings there soon came stories about giants and dragons that once inhabited the earth.
The ancient storytellers were right about the bones being those of giants, and the description of dragons is fairly close to that of dinosaurs. It would take time, scientific research, and the discovery of more specimens to dispel many of the myths surrounding the origins of these fossilized remains. Though it wasn't until the name "dinosaur" was applied to fossilized remains in 1841 that fossil hunters knew what they were looking for, the discovery of a large femur bone in 1676 by Reverend Plot in Sussex England pointed to a history of life on earth that we knew nothing about at that time. Mary Anning was another early fossil hunter and is credited with one of the earliest fossil discoveries, finding an ichthyosaur with her brother between the years of 1809 to 1811. In 1819 the Iguanodon was discovered by Gideon A. Mantell, but Mantell didn't name the creature until years later. Mantell's finding went unnamed until after William Buckland discovered and named the Megalosaurus, and was credited for finding the first scientifically named dinosaur.
Up to this point it had only been fragments of dinosaurs that had been found, and Gideon A. Mantell's discovery consisted of what was actually only a few teeth. Many specimens had been found though and some of which were even sold, but it was a discovery in the United States that yielded the first nearly complete dinosaur skeleton. In 1858 William Foulke earned himself the distinction of finding not only the first dinosaur in the United States, but the first complete dinosaur that had ever been found. Foulke's discovery in Haddonfield, New Jersey was given the name Hadrosaurus and with this find he inspired would-be fossil hunters and future paleontologist all over the world. Whether or not this was the first discovery of dinosaur remains is arguable, but it was this complete find that spurned so many young students to pursue archeology, and it furthered the science more than any other discovery that came before it.