Collections are a set of 'items' dropped by monsters (except dungeon monsters). Once all the pieces in the collection are obtained, the player can trade it in for a prize. Most collections offer magic stones, crafting ingredients or energy (food) items. However, all boss monsters' collections give a piece of clothing instead.
Monsters are not the only thing that drop collection items, reactors in your home also have collections. These give more interesting items such as energy, food, and permit items.
All collections can be found through the "My Stuff" menu with the completed ones listed first. Collection names hint which monster it belongs to (most are very obvious).