Tension/extension spring
The spring is designed to operate with a tension load, so the spring stretches as the load is applied to it.
Compression spring
Designed to operate with a compression load, so the spring gets shorter as the load is applied to it.
Torsion spring
Unlike the above types in which the load is an axial force, the load applied to a torsion spring is a torque or twisting force, and the end of the spring rotates through an angle as the load is applied.
Constant spring
Supported load remains the same throughout deflection cycle.
Variable spring
Resistance of the coil to load varies during compression.
Variable stiffness spring
Resistance of the coil to load can be dynamically varied for example by the control system, some types of these springs also vary their length thereby providing actuation capability as well.
Flat spring
Made of a flat spring steel.
Machined spring
Manufactured by machining bar stock with a lathe and/or milling operation rather than a coiling operation. Since it is machined, the spring may incorporate features in addition to the elastic element. Machined springs can be made in the typical load cases of compression/extension, torsion, etc.
Serpentine spring
A zig-zag of thick wire, often used in modern upholstery/furniture.
Garter spring
A coiled steel spring that is connected at each end to create a circular shape.