The links between modern day birds and their dinosaur predecessors are many. So what makes a bird a bird, and is trying to separate birds and dinosaurs really just a distinction without a difference?
In response to this post on a new on-screen T-Rex covered in a mixture of feathers and down, a discussion began about just how to distinguish a bird from a non-bird — and just where dinosaurs fell on that spectrum:
Birds can be defined as "a group of theropod dinosaurs which, ancestrally, are capable of flight". That word "ancestrally" covers the fact that some modern (and even extinct) birds are not able to fly. The issue is that when birds were first identified as a group, no one knew they were theropod dinosaurs. ... and, heck, there's people today who are contrarian about it.
So birds are one branch of the dinosaur family tree, which means that when people, in scientific speak, talk about dinosaurs, they should be including birds. Not all people do that, and in non-scientific speak, people talk about "dinosaurs" and birds as if they are somehow unrelated to one another. Which makes as much sense as talking about apes and humans like human's aren't a type of ape.
Dinosaria is a monophyletic group of all dinosaurs, including birds. Contained within that group are the "avian dinosaurs" which form a monophyletic group commonly known as birds. "Non avian dinosaurs" is a paraphyletic group that consists of all dinosaurs with the exception of birds. The line between bird and non-bird is blurry, and there is some discussion about where to draw it. But to call all coelurosaurids—dinosaurs with hollow bones and feathers—birds, or even bird-like is too inclusive. This clade includes T-rex, Compsognathus, Velociraptor, and the Oviraptor. Most members of this clade have descendants that are nothing like birds.
Calling eumaniraptora (also known as paraves) bird-like makes sense. This group includes both birds and dromaeosauridae (velociraptors and friends). But still, not birds, and not avian dinosaurs. Even Archaeopteryx, which is very birdlike and very closely related to birds, are on a fork that split off from the lineage that became birds. The Archaeopteryx branch failed, birds survived.
tl;dr: avian dinosaurs=birds. non-avian dinosaurs=all dinosaurs except birds.
Archaeopteryx: not a bird. Velociraptor: really not a bird. T-rex: really, really not a bird. Stegosaurus: so not a bird it is ridiculous. Dimetrodon: Not even a dinosaur.
Non-avian dinosaurs refers to all dinosaurs that aren't in the class Aves, and therefore aren't true birds. Many non-avian dinosaurs still had feathers though. Although, to be fair, the distinction between non-avian dinosaurs and true birds is a little fuzzy.
Image: Velociraptor mongoliensis / Matt Martyniuk