Very nice question !
Since we were not there, we do not know which exact dinosaur was last to die, but most scientists believe that a meteor hit Earth and wiped out ALL living dinosaurs that were on Earth at the time. So that includes all dinosaurs of various different species: triceratops, T-Rex, brontosaurus, etc..
Technically, all birds are types of dinosaurs. Birds evolved from dinosaurs, and according to modern systems of classification (cladistics) are therefore classified as a type of dinosaur. So the answer, is the birds – they are still alive today.
As far as non-avian dinosaurs are concerned, they went extinct 65 million years ago. Lots of dinosaur species (and many other animals) died out at the same time – and nobody is sure why, though an asteroid impact in the Yucatan peninsula in modern day Mexico is the leading hypothesis. Some of the last dinosaur species that we have found fossils for were Achelosaurus, Acrocanthosaurus and Agujaceratops. It is possible that some dinosaurs survived for longer than these, but without fossil evidence it is impossible to tell when the very last dinosaur passed away.