Here, gentle reader, all your questions will be answered...but first some background.
Anime is the Japanese word for "animation", and American fans (and others) use it to refer to Japanese animation. Anime tends to have a number of themes and conventions which don't appear in other types of works, that lend themselves well to fanfics.
"Fanfic" is short for "fan fiction". Fanfics are stories written by fans of certain types of works. Anime fanfics can feature the characters of existing anime in different situations, or they can simply feature an "anime flavor" (like the stories on this site).
ImproFanfic is short for "improvisational fan fiction". It's a web site, found at www.ImproFanfic.com, which features "impro" stories. The basic idea of impro is that writers take turns writing chapters (also called "parts") that continue the story. In more specific terms, if you want to write for an impro, you email the admin who assigns you a slot in the queue. Once the part before yours is posted, you have a set amount of time (usually around a week) to write the chapter, then send it in, again by email.
These questions are beyond the scope of this document. All you need to know to write for Beware the Radish is contained in the rules pages, the How is this site different from ImproFanfic? page, the other information pages...and, of course, the stories.