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)
               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.