The Thompson's books are divided by grade (first grade, second grade, etc.) Aside from that I have encountered a pianist who says that he is considered to be an X grade pianist. what is this grading system exactly?
There are various grading systems used throughout the world. Grades are set by various music organizations and are designed to reflect the difficulty of playing a song, or the skill depth and breadth of a musician.
In Canada, I took lessons from teachers certified by the Royal Conservatory of Music and followed the RCM's grading system, complete with songbooks of graded songs and examinations. (I got my grade 8 piano before stopping lessons).