Picarats are points that you receive after completing a puzzle in the Professor Layton series. These Picarats are used to indicate the difficulty of a puzzle, with the easiest starting at 10, and the most difficult ending at 99. The only use for Picarats is for unlocking bonus features in the Top Secret section of each game, such as character profiles and cut scenes.
When you fail a puzzle, the value of Picarats the puzzle is worth goes down, usually by 10%. This does not, however, reduce the difficulty but instead gives you less Picarats when you successfully finish it. The Picarats will only reduce by amount twice, any failures after that and the Picarat total will remain the same no matter how many times a the player fails a puzzle.