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Leaf Springs
Leaf springs are also called laminated, semi-elleptical, carriage springs, cart spring, or helper spring. These are the simplest form of spring, commonly used for the suspension in vehicles.
These are an example of one of the oldest forms of spring making, dating back to medieval times. This type of spring is made from flat spring steel of rectangular cross-section. The steel is formed into an arc. The center of the arc provides the location for the axle, while tie holes are provided at either end for attaching to the vehicle body. For very heavy vehicles it is normal for several leaves to be stacked on top of each other in several layers, often with systematically shorter leaves. These springs were very common on automobiles up the mid 1970's. Now most designs of automobile manufacturers use coil springs, gas springs, or air suspension instead. They are still used in heavy commercial vehicles such as vans and trucks, and rail cars. For heavy vehicles, they have the advantage of spreading the load more widely over the vehicle's chassis, whereas coil springs transfer it to a single point. Unlike coil springs, these springs also locate the rear axle, eliminating the need for trailing arms and a Panhard rod, thereby saving cost and weight in a simple live axle rear suspension.
Elliptical or full elliptical springs refer to two circular arcs linked at their tips. These are joined to the frame at the top center of the upper arc. The bottom center is joined to the suspension components, such as a solid front axle. Additional suspension components, such as trailing arms, are needed for this design. Semi-elliptical springs use a lower arc. Therefore, they don't need the additional suspension components. Quarter-elliptical springs have the thickest part of the stack of leaves stuck into the rear end of the side pieces of a ladder type frame. The free end is attached to the differential. An example of non-elliptical springs of this type is the Ford Model T. It had multiple springs over its differential that were curved in the shape of a yoke. As a substitute for dampers (shock absorbers), some manufacturers laid wood sheets in between the metal leaves. 
A newer design is the parabolic leaf spring. This design uses fewer leaves. The thickness of the leaves vary from center to end following a parabolic curve. Friction between the leaves is not wanted. So there is only contact between the springs at the ends and at the center where the axle is connected. Spacers prevent contact at other points. The advantages of this type design are weight saving and greater flexibility. They can come very close to the same performance of coil springs. Rather than repeat what has been done...Here are a couple of good web pages on leaf spring design.
Helper Spring Design
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