Design Mode is reached by clicking on the four-way arrow just to the left of the Marketplace button. There are five things you can do in Design Mode: Move animals, buildings, and objects; Rotate buildings and objects; Sell animals, buildings and objects; Paint buildings; and Store animals, buildings and objects. A storage cellar will increase your storage capacity. Some special buildings you can unlock will allow you to store animals in them.
Design Mode is disabled while visiting Neighbors, the Fairy Realm, or Land's End.