The five classes of vertebrates are fish, amphibians, reptiles, mammals and birds. These groupings predate modern genetics and represent functional groupings rather than true trees of phylogeny. Bony fish, for example, are more closely related to mammals than they are to sharks.
Vertebrates are members of a sub-phylum within the phylum Chordata. The five vertebrate classes are grouped according to classical taxonomic lines rather than by clade, which is a true representation of each group's real relationship with the other groups of chordates. A clade consists of all descendants of a common ancestor. Thus, mammals and birds are good clades, as each group is descended from its own common ancestor, which was not ancestral to any other non-mammal or non-bird. Reptiles are a grade, not a clade, as some reptiles are more closely related to birds than they are to other reptiles. The same is true of amphibians, who are also not a true clade, as some descendants of their common ancestors are now reptiles, birds or mammals.
The four classes of air-breathing vertebrates are called tetrapods, and they have close relatives among bony fish. Tetrapods and bony fish are a good clade, as they are all exactly equally distant from the vertebrate outgroup of cartilaginous fish.