How are birds organized, scientifically? What phylum, subphylum and class do they belong to? What makes a bird a bird?

KINGDOM - Animalia (i.e. animals, not plants or bacteria etc)
PHYLUM - Chordata (i.e. with a nervous system)
SUBPHYLUM - Vertebrata (i.e. with a main backbone)
CLASS - Aves (i.e. having feathers)

So all birds fall into the Aves class. Whether they use those feathers to fly or not, if they have feathers, they are a bird.

There are four groupings below the class level, to help sort out the bird groupings. These are:

ORDER
About 28 orders exist, and these hold large gropings of similar birds, such as owls, pigeons, penguins, etc. These are the main groupings of birds. The order Falconiformes holds all the falcons, hawks and eagles. The order Anseriformes holds swans, ducks and geese.

FAMILY
A family of birds is within a given order. They help further break down an order. So while an order might include "large flightless birds", a family might include "emus".

GENUS
A genus is the second smallest level of distinction and identifies a very small groping of birds. The genus name is always shown in the two part latin name of a bird. So for example, chickadees would be a genus while specific types of chickadees (boreal, mountain, etc) would be the species.

SPECIES
The species is the smallest unit of distinction for any living creature. For birds, this defines the smallest group of birds that still has defineable characteristics - breeding within its species, living in a certain type of habitat, having similar songs, eating similar foods. New species are formed when a group of birds somehow gets isolated from the rest of its fellow birds and begins to learn to live under new conditions, eating new foods, developing new songs.

Scientific Definition of a Bird - General Overview
Bird Classification Example - Chickadee
The Orders of Aves Birds