Depending on what kind of 'dragon' you're talking about, the fantasy varieties are just that. Otherwise the Komodo dragon is very much alive today.
Species go extinct all the time- there are hundreds and thousands of them, insects, animals, plants alike, that we aren't even aware of, and might never be. This is part of a natural turnover due to competition between species- and the debate largely centers on how responsible we humans are for shifting the balance of such things.
Some other notable extinct species include the dodo (a flightless, rather daft bird hunted to extinction for its meat), and the thycaline (better known as the Tasmanian tiger, also overhunted to extinction).
Heavens, crazy, open an encyclopaedia of animals or something!
I think we kinda moved onto talking about endangered species here, of which there are thousands that we keep an eye on, including elephants, giant pandas, snow leopards and several species of whale. Because if they die out, then they become extinct!
Most of the species that lived in the prehistoric era, like dinosaurs, sabre-toothed tigers, wooly mammoths, either evolved or have become extinct, yes.
Coelocanth! thought to be extinct thousands of years ago, but it showed up off the coast of Madagascar! If we're lucky (we're not) some of the extinct animals are just "hiding". Every so often there is a thylacine siting...
On endangered species, seems like the biggest threat is habitat destruction. It affects all animals at once, and it moves at a rapid pace.
And finally, some animals (like Strop noted) are yet to be found! A small mammal was found on Borneo few years ago, not sure if it was identified yet; and a known leopard was discovered to be a new species (GO BORNEO!)
