Blue Temples are the only place you can find a Blue Dragon Egg.
They are made of Blue Bricks in a rectangular shape with a glass roof. The floor is edged with water one block wide inside.
The entrance is in the center of the roof in the form of a slot where water falls down to a hole in the temple floor.
Blue Temples are found in Ocean Biomes, on the floor of the ocean.
The Blue Dragon Egg requires nothing special to get, it sits on top of glowstone inside the temple.