So a Villanelle is a form of poetry from France. In it, there are Tercets, which are stanzas (the little paragraphs in poems) of three sentences. There are 5 of them in a Villanelle.
Now, the final stanza (not one of the Tercets) is called the Quatrain.
The Quatrain the first and third lines tie together. The basically make up the heart of the poem.