Chordates (phylum Chordata) include the vertebrates, together with several closely related invertebrates. They are united by having, at some stage in their life, a notochord, a hollow dorsal nerve cord, pharyngeal slits and a muscular tail extending past the anus.
The phylum Chordata is broken down into three subphyla: Urochordata, Cephalochordata, and Vertebrata. Urochordate larvae have a notochord and a nerve cord but they are lost in adulthood. Cephalochordates have a notochord and a nerve cord but no vertebrae. In vertebrates, the notochord has been replaced by a bony vertebral column.
The extant groups of chordates are related as shown in the phylogenetic tree, below. They do not match up very well with the traditional groups, and as a result vertebrate classification is in a state of flux, although their relationships are very well understood.
Note that snakes are considered tetrapods because they are descended from ancestors who had a full complement of limbs. Similar considerations