Max's dream journal is a leather-bound journal Max uses to record his dreams in, specifically the ones about Sharkboy, Lavagirl, and Planet Drool. The book is well-worn and full of Max's dreams, imaginative thoughts, and general doodles. Due to the amount of effort he put in it and valuable information (to Max) in it, Max is very protective of the journal.
Since the journal is so valuable to Max, Linus made it his goal to steal it from him and vandalize it. When he succeeded with his mission, he scribbled with a black marker over (seemingly) every page in Max's journal.
While on Planet Drool, the journal was in Minus' possession and undamaged.
Max temporarily got it back only for Lavagirl to accidentally burn it to ashes when she held it in her hands. Later in the movie, in reality/on Earth, it's struck with electricity and charred beyond recognition/use. At this point Max had matured enough to understand that he'd have more dreams and didn't need to rely on the journal anymore so he discarded it without a second thought.