The short answer is yes
Every bird alive today, from the robin on your lawn to an ostrich or a penguin, is a dinosaur. Birds came from small, feathered dinosaurs that walked on two legs, in the same big family as Velociraptor and T. rex. When the age of dinosaurs ended, the birds were the dinosaurs that were left.
So when someone says the dinosaurs are gone, you can answer: the ones that were not birds are gone. The rest are singing outside your window.
What birds share with their dinosaur family
Feathers: fossils show that many dinosaurs had feathers, long before any bird flew. Velociraptor's arm bones have little bumps where feathers once held on, so we know it wore them, even though it could not fly.
Hollow bones and a wishbone: a bird's bones have air inside, which keeps them light. Many dinosaurs had light, hollow bones too, like the speedy Gallimimus. The wishbone in a chicken or a turkey is a pair of collarbones joined into a V, and many meat-eating dinosaurs, including Velociraptor, had one as well.
Eggs in nests: birds lay eggs and sit on them to keep them warm. Fossils have been found of dinosaurs sitting on nests of eggs in just the same way.
Look at a bird's legs
Next time you see a pigeon or a robin, look down at its legs and feet. They are covered in scales, with claws at the end of the toes, and the bird walks on two legs, just as its dinosaur family did. You are looking at a real living dinosaur.
Not every flying animal was a dinosaur
Pteranodon and Quetzalcoatlus flew above the dinosaurs, but they were pterosaurs, a different group of flying reptiles. A pterosaur's wing was skin stretched from one very long finger to its legs. A bird's wing is feathers on arm bones. Pterosaurs lived at the same time as the dinosaurs and were their cousins, not their parents.
Sea reptiles such as the long-necked plesiosaurs shared the world with the dinosaurs too, and they were not dinosaurs either. Birds came from dinosaurs, not from pterosaurs.
For Big readers: where birds sit on the family tree
Scientists sort living things by family tree, and a group on the tree includes everything that grew out of its starting point. Birds branch off from inside the meat-eating dinosaur group called theropods. That makes birds a group inside the dinosaurs, the way a poodle is a group inside the dogs. It would be wrong to ask whether a bird is "related to" a dinosaur, the way you might ask about a cousin: a bird is one.
Fossils show the steps. A famous fossil called Archaeopteryx, found in Germany, has feathered wings like a bird, but it also has teeth, claws on its fingers and a long bony tail, like its dinosaur relatives. Since then, fossils from China have shown many more feathered dinosaurs in between. Exactly which of them could fly is something scientists still argue about.
Pick Big at the top to see the part for older readers.