I know there's a better explanation elsewhere, but here's the basic gist;
A group of people take turns to provide a section (usually around a minute) towards an eventual piece made up from the different contributions placed together to make a whole...
The catch is that each contributor only gets around ten seconds of the end of the predecessor's section to link their section from.
In a Lyrics-First Corpse, it's slightly different, wherein a group of lyricists provide different sections, until all of the sections have been completed, once again passing on the last snippet of their sections to the next person.
Once those lyrical bits are finished, then the musicians/singers for them to music, once again in a similar fashion.
LYRICS FIRST EXQUISITE CORPSE 1 - *18637
1. @Amanda West
5. @Stephen Wordsmith
2. @Vom Vorton
4. @Rob From Amersfoort