First off, let us clarify - since birds are dinosaurs, I am going to assume your question is really “when was the first non-avian dinosaur found?” Otherwise, the answer is the first dinosaur was found the first time a human saw a bird!

By the way, if you want to pursue why birds are considered as a derived group within the dinosaurs, an excellent resource is the University of California Museum of Paleontology page on this subject

So, what is the history of the discovery of the large terrestrial dinosaurs, the extinct lineages of Ornithischia and the non-avian Saurischia? In one sense, the answer to that question is difficult, as historical records indicate that dinosaur bones were recovered long before Richard Owen recognized dinosaurs as a distinct group or branch of life in 1842.

Indeed there are plausible descriptions in the Chinese literature of 2,000 years ago of bones recovered from the Earth that were supposed to be those of dragons, but which might well have been those of dinosaurs. Similarly, in 1677 an English Historian, Robert Plot, described a huge bone from a quarry in Oxfordshire, England that was probably a dinosaur bone, although it has since been lost.

The first recognition of a large, extinct reptilian animal of the Mesozoic was offered by William Buckland in 1824, when he described a lower jaw with teeth as “Megalosaurus” and realized that it was a large reptile – a good summary is given at this link.

Several more English discoveries followed – by example Iguandodon was recognized 1825 by Gideon Mantell (The Wikipedia entry on “Discovery and History” of Iguanodon similarly gives a good summary of this discovery.

Again, these were not called “dinosaurs” until the Dinosauria was defined as a distinct lineage in 1842, although the clear link between birds and dinosaurs was not fully accepted until the latter portion of the 20th century.

Fossils that would now be considered dinosaurs have been known for centuries, but early scientists/natural historians had little idea what they were looking at. Not until 1841 did Sir Richard Owen coin the term "dinosaur," a term he applied to three species of extinct reptiles known at the time. He is thus the first person to recognize dinosaurs as a distinctive group of animals.

Dinosaur fossils have been known since prehistoric times. More than a few mythical monsters from gryphons to dragons have probably been inspired by dinosaur fossils.

The OLDEST dinosaur fossil currently known is from the early Triassic period, between 230 and 240 million years ago.