Basically, it starts with a challenge. I'll use an example that happened where I live.
A mother didn't approve of her daughter reading the book "Speak" due to its content. She brought the book to the Principal and explained her stance on it. He brought it to the administrators.
Here it can go either way. If the woman is not justified, then they can find an alternate book for her student to read, while the rest of the class reads the book. That's what happened with "Speak" thankfully, however it's not always that pleasant.
If the administrators feel she IS justified, they need to bring it to the school board. There it's discussed and voted upon.
A book isn't necessarily banned from EVERYWHERE, it can banned from a single school or from an entire district. It depends on the outcome of the school board meeting.
If it is banned, it simply means that it cannot be found in the school library nor taught.
A lot of good books have been banned. Thankfully people are standing up against it. Books like "The Adventures of Huckleberry Finn" should still be taught, regarldess of language. And child books like "Are you there God, it's me Margaret" are very unjustly banned.
Sorry, I'm slightly passionate about this subject.
Hope that helps!