Blue coins are not simply coins that are blue; for every ten that you collect, a Shine Sprite is waiting as your reward. There are many blue coins spread out around the game; you'll find them inside zones and out in the streets of Delfino. There are 240 blue coins to be found throughout the game.
Take your hard earned collection to the bear in the boathouse (the portal to Ricco Harbor) to buy Shines with them. You'll need 10 Blue Coins for a single Shine.
Head to one of the following pages for more information on how many blue coins there are in that area and how to acquire them.