A poem — especially one that rhymes — is called verse. The children's author Dr. Seuss wrote in verse, and the regular rhymes of "The Cat in the Hat" helped generations of children learn to read.
In music, a verse is a section of a song that's often followed by a chorus. Many people can sing the chorus to "Take Me Out to the Ball Game," but few realize there are also verses, including one that starts, "Katie Casey was baseball mad." Verse comes from the Latin word versus, which means a line of writing and is based on the Proto-Indo-European root wer, meaning to turn or bend.