Chordata

The traditional classification of vertebrates is a mess, with a wide variety of paraphyletic groups that have either been abandoned or extended (here marked B). Unfortunately, no particularly standard system has settled in to replace it; the problem is there have been too many adaptive radiations. For now, here is a minimal tree of living forms including most traditionally class-sized groups:

Urochordata?

Cephalochordata?

Craniata?

Myxini? (hagfish)

Vertebrata?

Petromyzontiformes? (lampreys)

Gnathostoma?

Chondrichthyes (cartiligenous fish)

Osteichthyes B

Actinopterygii (ray-finned fish)

Sarcopterygii? B

Coelocanthamorpha?

Dipnoi? (lungfish)

Tetrapoda?

Lissamphibia? (frogs, toads, salamanders)

Amniota?

Mammalia

Chelonia? (turtles)

Squamata? (lizards, snakes)

Sphenodontida? (tuataras)

Crocodylomorpha? (crocodiles, alligators)

Dinosauria: Aves (birds)

Information on characteristics of each group and a more detailed classification thereof should probably go under its headline - eg details of skeletal system and listing of jawless fishes under Vertebrata.

 Subphylum Urochordata (Tunicata)
     Class Ascidiacea?
     Class Thaliacea?
     Class Appendicularia? or Larvacea?
 Subphylum Cephalochordata
     Class Cephalochordata? or Acraniata? or Leptocardia? (Lancets)
 Subphylum Vertebrata
   Superclass Agnatha ("Without jaws")
     Class Ostracodermi?
     Class Cyclostomata?
     Class Myxini?
     Class Cephalaspidomorpha?
   Superclass Gnathostomata ("Jawed mouth")
     Class Placodermi? (an extinct fish)
     Class Chondrichthyes (cartilaginous fishes)
     Class Osteichthyes (bony fishes)
     Class Amphibia or Anamnia? (amphibians)
     Class Reptilia (reptiles)
     Class Aves or Sauropsida? (birds)
     Class Mammalia (mammals)