Snails are invertebrates—animals without a backbone (vertebral column). They are mollusks in the class Gastropoda: soft-bodied animals that usually carry a single coiled mineral shell (slugs are closely related gastropods that lack a visible shell).
Vertebrates are animals with vertebrae, such as fish, amphibians, reptiles, birds, and mammals. Snails do not have that internal bony or cartilaginous spine, so they are classified with other invertebrates.
Source: Kentucky Geological Survey, University of Kentucky — Gastropoda (snails).