First let me correct a misconception. Birds do not appear earlier in the fossil record than dinosaurs. They do, however, overlap dinosaurs by quite a lot, which is a point some creationists use as "proof" that birds cannot have evolved from dinosaurs, though they look over the fact that this would not say they didn't evolve at all!
I am sure you can see the fallacy of such an argumnet... it's just another version of the old "If humans evolved from apes, why are there still apes?" argument.
Our knowlwdge about the evolution of birds from therapod dinosaurs is still very young... and we're learning more on an almost monthly basis.
The deabte about the exact evolutionary origin of birds is still fairly heated. Many argue that the Theropods are the ancestors of modern birds and that the earliest birds show clear evidence of such a relationship.... for example, scientists now know archaeoptyrex did not have an opposed rear toe, so it's feet looked exactly like a dinosaur's foot and not like a modern bird's foot.
In fact, although archaeoptyrex has usually been considered a dinosaur-like bird, some now consider it a bird-like dinosaur.
Of course, we know about feathered dinosaurs, but the issue does get murkier.
The earliest "birds" seem to appear earlier in the fossil record than the best examples of unambigusously feathered dinosaurs, though cladisitc research indicates that many of the earlier fossil theropods we do have were probably feathered, but we just don't have the skin impressions to prove it. A small minority think the poorly preserved fossil of Longisquama insignis is evidence that birds and dinosaurs had a common ancestor, rather than birds descending from dinosaurs themselves. Those that are proponents of this idea hold that "feathered dinosaurs" are actually early flightless birds, though this idea has it's own complications.
Of course, this gets to your main question, which is one of classification. It's important to realize that classification is a tool invented by humans, and not imposed by nature. We find classifcation useful as a way of describing relatedness, but nature is not at all concerned about defining precise boundaries to assist us. And it only gets more complicated as we find more fossils which fill in the gaps of the record. So, is archaeoptyrex a bird-like dinosaur, or a dinosaur-like bird? It certainly has some features we simply don't usually associate with modern birds. But it is certainly more bird-like than what we tend to think of as a dinosaur.
So the short answer, I think, is that scientists are finding it difficult to draw a bright line between dinosaurs and birds. If you polled 10 paleontologists about this subject, you probably find 8 different answers and 2 who say it doesn't matter.
They do, however, overlap dinosaurs by quite a lot, which is a point some creationists use as "proof" that birds cannot have evolved from dinosaurs, though they look over the fact that this would not say they didn't evolve at all!
Even worse, they still could've evolved from dinosaurs even if they are overlapping.
We should also point out that there are examples today of a trait like flight evolving, being lost, and being reacquire so it is entirely possible that dinosaurs could have evolved into flying birds that then evolved into flightless birds.Anyone so limited that they can only spell a word one way is severely handicapped!
Currently, a bird is Archaeopteryx or anything that is more like a modern bird than Archaeopteryx is.
More accurately, 'bird' is not a scientific term, but a common one, and doesn't have any clear scientific definition. From an eviolutionary point of view, recognising that change is gradual, it's not too important where the exact line is between protobird and bird.
In scientific nomenclature, the birds are 'Aves', and the most common definition of Aves is the one that Dr. Adequate says - the common ancestor of Archaeopteryx and modern birds and all its descendants. This isn't the only definition though, and to try and clarify issues like this and give us clear, unambiguous definitions of groups of organisms, a mighty project is under way called the Phylocode. This will give a standard definition for each taxon so that, in theory at least, anyone working in biology would be able to check exactly what you do and don't mean by 'Aves' (when it will ever be ready is an open question).
Under some of the proposed definitions, Archaeopteryx wouldn't make it into Aves, but whether or not you wanted to carry on calling it an early bird is a matter of taste. Common terms for animals always hit problems when you go back towards distant ancestors, that's why some bioogists decided they needed this unambiguous series of definitions for scientific nomenclature.
There are many differences between birds and dinosaurs. Their lungs, reproductive system, etc. Birds breathe more because they have special lungs. They're incapable of moving their thigh bones like that of dinosaurs, so their lungs will collapsed if they do.