Doves are small white birds that are usually thought to bring happiness and love. For those reasons, they are commonly released at weddings. They are also considered a symbol of peace.
Shadow of the Colossus
In Shadow of the Colossus, doves surround Mono. This is, perhaps, to symbolize her purity. They may also have plot-related functions; as each colossus is felled, another bright white dove appears at Mono's altar, matching the number of defeated colossi. This matches the number of black shadows that appear around Wander's body and the pillars of light, leading to much fan speculation on their purpose and meaning. These doves are absent from the ending until the final scene in the garden where everything returns to peace. This may tie back to their use as a symbol of peace and serenity.
They don't do anything to harm Wander, and he can even take a brief flight on one of their wings if he catches one before it flies out of reach. Some doves live in the Forbidden Land, usually collecting around water. They are easily spooked.
Doves are also seen in some areas of the castle in Ico. If the player remains in the same place without doing something important (fighting, solving puzzles) Yorda might chase them for fun. The doves serve no function except for creating atmosphere.