A story in verse

One word substitute is Ballad.

Elegy: (in modern literature) a poem of serious reflection, typically a lament for the dead.
Hymn: a religious song or poem of praise to God or a god.
Sonnet: a poem of fourteen lines using any of a number of formal rhyme schemes, in English typically having ten syllables per line.
Ballad: a poem or song narrating a story in short stanzas. Traditional ballads are typically of unknown authorship, having been passed on orally from one generation to the next.