The rondine is a French verse form from around the 16th century that traces its roots to the rondeau, joining a prestigious family of forms that are united by difficult rhyme schemes and heavy use of refrains. The rondine, for its part, is effectively just a shorter rondeau, at 12 lines instead of the usual 15.
The rondeau redouble is essentially a challenging “super rondeau” that extends the form out to 25 lines instead of 15. That, while also demands a tighter structure focused around rhymed quatrains instead of uneven verses, while still utilizing refrains. It was invented by the 16th-century French poet Clement Marot.
Huitains are eight-line poems that were mostly popular in 16th century Western Europe. They are usually credited to France and generally consist of the first eight lines of the longer ballade form, though there has been some historical experimentation with the rhyme scheme and line lengths of the huitain.