A quatrain is simply a poem with 4 lines.
The rhymed quatrains of Nostradamus were written mainly in French with a bit of Italian, Greek, and Latin thrown in. He intentionally obscured the quatrains through the use of symbolism and metaphor, as well as by making changes to proper names by swapping, adding or removing letters. The obscuration is claimed to have been done to avoid his being tried as a magician. Of course a skeptic might say it was done so the quatrains could be interpreted to fit numerous situations.
An example of one of his more famous quatrains is from Century #2 Quatrain #24. The French and English versions are as follows:
Bestes farouches de faim fleuues tranner;
Plus part du champ encontre Hister sera,
En cage de fer le grand fera treisner,
Quand rien enfant de Germain obseruera.
Beasts ferocious from hunger will swim across rivers:
The greater part of the region will be against the Hister,
The great one will cause it to be dragged in an iron cage,
When the German child will observe nothing.
So does Hister actually refer to Hitler? An instance of Nostradamus changing letters around? Is this a reference to Germany during WWII?
If you'd like to read and search through all of Nostradamus' Centuries visit the Nostradamus Prophecy Search page.