The title most obviously refers to Liesel Meminger, the chief book thief of the story. She's officially given the title by her best friend Rudy Steiner at the end of Chapter 42. In this chapter, Liesel steals The Whistler from Ilsa Hermann's library, the first of many such library raids. But, since Rudy is her accomplice, isn't he a book thief, too? Come to think of it, he's more than just an accomplice. He actual re-steals The Whistler from the Amper River, where the second leader of the fruit stealing gang has thrown it after stealing it from Liesel (who stole it from Ilsa Herman).
Death, our narrator is also a book thief. When Liesel drops her newly completed memoir, The Book Thief, after learning that all those she knows and loves on Himmel Street have died from bomb blasts, Death steals the book from a trash truck.
And then there's Max Vandenburg, the Jewish man hiding from the Nazis, who commits an ultimate act of figurative book thievery. He paints the pages of Adolph Hitler's book, Mein Kampf, white and then sketches loving and frightening words and pictures of his life over the paint.
Hitler, while we're on the topic, is actually the biggest book thief of the novel. This becomes apparent in Part 2, which features a massive book burning in celebration of Hitler's birthday in 1940. (You can watch footage of an actual Nazi book burning here.) Of course, this is no isolated incident. Book burnings occurred throughout Germany, throughout Hitler's reign. Destroying the books was symbolic of the desire to destroy the people who wrote the books and the people represented in the books. Hitler's targets were primarily Jewish people, but he also went after communists, disabled people, homosexual people, Jehovah's Witnesses, Gypsies and others considered "undesirable" by the Nazi regime.
With every act of book-thievery, Liesel and her crew steal back whatever words and books they can from Hitler, and even steal his book from him (though he doesn't know it!). Although this doesn't (arguably) make a huge difference in the grand scheme of things, it makes a huge difference in the lives of these tortured characters. The novel cautions us – don't take your books for granted. Someone could steal them from you at any moment. And if someone does steal your books, will you be ready?