This item is used to store your many medals collected throughout your journey - Source What the medal box is
Medals are a new feature introduced in Pokémon Black 2 & White 2. These medals are given to you to commemorate various different accomplishments throughout the game. These accomplishments vary in the means to obtain them, from catching Pokémon to doing specific tasks.
To receive the medals, you just need to speak to the Medal Master who sits in each of the Pokémon Centres. If he has any medals to give you, you will receive them all in one go. -Source How to get medals