Feathers are awarded for achieving a 100% Total Destruction score in any level. To do this you must use the Mighty Eagle to destroy as many blocks in the level as possible. The Mighty Eagle can be activated by clicking the Sardine Can button from the power-up menu in the story levels.
In order to maximize your score, it is recommended that you first fling all of your birds and then use the Mighty Eagle afterwards. It is almost impossible to get a 100% Total Destruction feather in most levels by using only the Mighty Eagle.