What kind of vertebrate is an eel?

Yes, all fish of any kind are vertabrates belonging to the phylum chordata. Which means the animal has a spine and spinal chord. Humans also belong to this phylum of animals. Now when we start talking about shrimp and lobsters and such the answer becomes no, invertabrates have no spine and derive their shape from their exo-skeleton. Not all invertabrates however belong to the same phylum.