I apologize in advance for anyone who wasn't aware that birds are technically dinosaurs. But then again most of us who have heard this little fact somewhere haven't ever really bothered to digest enough to consider why birds are classified as dinosaurs. Although, to be fair, it really isn't something very important to daily life.
Before I get into the technical side, I'd also like to force you to consider the implication of birds being related to dinosaurs: birds are reptiles. Yes, birds are under the same classification as one of those little geckos. They aren't in a separate classification like mammals.
Enough of that though, so why are birds considered dinosaurs? To start as confusingly as possible, there are two types of dinosaurs: Sauriscian("lizard-hipped") and Ornithiscian("bird-hipped").
Saurischian(Lizard) Hip upload.wikimedia.org
Ornithischian(Bird) Hip upload.wikimedia.org
Here is where you are supposed to think that the bird hip is a... a bird hip? Well sorry but science sucks and that is actually wrong. As it turns out birds are descended from the lizard hip dinosaurs and the bird hip dinosaurs are all extinct. The reason behind that is a whole other story so just take my word for it.
So birds have a lizard hip, what does that mean? Well that means they are directly descended from such fan-favorites as Tyrannosaurus Rex and Velociraptor. Velociraptor even had feathers on its arms and legs and some evidence suggests T. Rex babies may have had downy feathers.
The Velociraptor is in the group most closely connected to birds, the Maniraptors. These include a few other examples but they are all bipedal carnivores with feathers and long tails. Eventually though we get to Archaeopteryx, the first known bird. Although this fellow is missing several characteristics universal to modern bird.
Archaeopteryx chasing down prey. upload.wikimedia.org
The major distinctions that make this "bird" seem off is the presence of teeth and a tail. Birds have a much more clearly defined beak without any teeth, but Archaeopteryx doesn't really have a beak like modern birds do. However this difference isn't too distinct, or at least as distinct as you might think.
There is actually evidence that it only takes a minor mutation for a bird, like a chicken, to gain teeth in its beak. If you want to learn more about that check out this article: https://www.scientificamerican.com/article/mutant-chicken-grows-alli/.
And for Archaeopteryx's tail, well that clearly needed to disappear, but some modern birds have tails. The roadrunner for example actually has a tail and still functions perfectly well.
Although tails are not very effective for flying which is why both the roadrunner and Archaeopteryx are poor fliers. But with enough time, Archaeopteryx's ancestors would have lost their tails to fly more efficiently and would have gained a more defined V-shaped wishbone which allows for modern birds' flapping motion.
So there you have it. Birds are dinosaurs because later dinosaurs were the ones to develop the characteristics that are present in birds today. And of course, birds are reptiles too. Thanks for reading!