Verse is a writing that uses meter as its primary organisational mode, as opposed to prose, which uses grammatical and discoursal units like sentences and paragraphs. Verse may also use rhyme and other technical devices that are often associated with poetry.
However, while much poetry is written in verse, not all of it is (see prose poetry)
Equally, not all verse is poetry. Generally speaking, what separates the two is that in poetry language achieves the highest possible level of condensation.
A verse is another name for a stanza.