Amy F. answered 09/06/19
Experienced Physiology & Biosciences PhD
The short answer is: The cytoskeleton.
Cells have structural proteins composed primarily of actin fibers. These fibers are bound to structural proteins embedded in the membrane, forming a web or skeleton that holds the shape of the cell. The placement of the membrane bound elements and the lengths of the actin fibers are all genetically determined for the cell type, so that each cell forms the appropriate shape. In some cases, chemotactic agents help to determine the length of processes off the membrane (especially in the case of nerve cell axons).