You probably noticed on the Radcliffe list of banned books in the articles section, every book also listed a reason the book was either challenged for a ban or banned.
Anyone can challenge a book. According to Google's Banned Book Week pages and the ALA, the most challenges to books come from parents. When you challenge a book, you file a formal complaint with your school, church, library, book store, and even local or state governmnet offices.
The challenge is recorded for posterity, and if the governing body of the location in which you filed decides your challenge has enough merit (or would cause too much of a politcal stink to dismiss) you can present your challenge formally.
The plus of living in a democracy is that concerned parents, librarians, teachers, politicians and civilians can fight your challenge. This results in many more challenges being filed than books actually being banned.
While it is true that more books are challenged than are actually banned, the issue of banning (and burning) books still rears its ugly head on a regular basis. If a book is actually banned, the usual recourse is to remove all copies of it from the location the challenge was filed.
If you know of a challenge being filed in your area and you would like to report it to the ALA and get help and advice in fighting it, please visit this link. All reports are confidential.