Because birds evolved from theropod dinosaurs in the Jurassic Period they are classified as dinosaurs because the modern taxonomy system that requires that all descendants of a single common ancestor must be included in a natural group. Thats why "dinosaur" is a natural group but "fish" for example is not. It is just a word to refer all the backboned animals that are not tetrapods (four-legged backboned animals). The word non-anian dinosaur is used to refer the dinosaurs that are not birds, though many people that are not scientists usually think only non-avian dinosaurs when they hear the word "dinosaur", but in scientific sense birds are a type of dinosaur.
Birds are the only dinosaurs that didn't went extinct. Though many of them died out with other dinosaurs. Birds are in the clade Avialae ("bird wings") of theropod dinosaurs.
It is believed that Deinonychosaurs dinosaurs, which include Dromaeosaurs (raptors) (like Deinonychus and Velociraptor) and troodontids (like Troodon), are the closest relatives of birds. It is sometimes hard to tell is a bird-like dinosaur, like Archaeopteryx or Microraptor an actual bird or not.
Dodo birds, like non-avian dinosaurs, are extinct. But, unlike non-avian dinosaurs dodos did live at the same time as humans.