Animals usually do not form multiple identical organs and usually can not grow back whole organs once they have been lost. Plants, however, usually can do this; they can produce thousands of leaves and roots and hundreds of stems in a lifetime. This process in plants is known as iterative development. It is important for plants to be able to do this becaue they have a transitory existence - their leaves and flowers fall off every year, their roots die and regrow, etc. This is a major difference between plants and animals.
In plants, new organs form at the meristems. This region of the plant contains undifferentiated cells called meristematic cells.They are regions where cell division occurs in the plant. An apical meristem is a meristem that occurs at the tips of the plant, and cause primary growth by lengthening the plant. Lateral meristems cause secondary growth by widening the plant.
A developing root or shoot tip normally contains four regions. The bottom-most layer is where the mature cells are - tisues in this region are usually fixed unless there is injury to the area. The next layer in the Zone of Cell Differentiation. In this region, mature cells start forming. The layer on top of that layer is the Zone of Cell Elongation. Here, cells elongate, with little division. The outer-most layer is the Zone of Cell Division - new organs begin to form in this layer.
After a cataclysmic injury, formerly fixed tissue can begin to regenerate and produce new shoots. This is due to plant cells' totipotent nature - they can de-differentiate, begin to divide, and re-differentiate into any other cell type at any point. This means that an entire plant could potentially be regenerated from a single plant cell.