Moshi Medals are numerous awards given to you in playing Moshi Monsters. In the agenda book / album there are a variety of medals according to each category of game that has you trying to accomplish.
Go to your room then see the ‘Medals button’ at left side just above the mood indicator. To collect all moshi medals, point your mouse into a box and follow the instructions on this medal book. If you’ve completed the mission automatically get a new medal by category.