In the mountain biking world, full suspension (or dual suspension) refers to a bicycle with suspension units on both the front and rear wheels.
Front Suspension Overview
On new production mountain bikes most front suspension units are telescoping forks, and are OEM sourced from either Rock Shox, Marzocchi or Manitou.
There are other fork manufacturers but they are either boutique manufacturers such as Fox or Pace who make forks for serious riders who know they are buying or any one of a number of cheaply made units which are mostly not suitable for serious off road riding.
The most common type of telescoping forks are 'normal' forks where the lower legs slide on the outside of stanchions joined to the crown and are joined together by an arch which ensures that the leg movement is synchronous. This is the kind of suspension fork you will see on almost any bike riding around town.
The other type of telescoping fork is the upside down fork where the lower legs retreat into the upper legs. The design is stiffer, but heavier and more expensive and also requires the use of disk brakes. The design is now only used for DH forks which have a fork travel of more than 150mm and require the stiffness of this design. The Marzocchi Shiver or the Cannondale Lefty are good examples of this method. You are more likely to see this design on a motor bike.
The crown is the element on the forks which joins the legs to the steerer and comes in two kinds.
A single crown which is the most common type, has a single piece of metal at the base of the headtube which the legs are attached to. It is seen on most bikes with any kind of suspension fork.
A double crown or triple clamp design has a crown at both the top and the bottom of the headtube and the legs pass thru the lower crown to join the upper crown. The triple clamp expression comes from the fact that the fork legs are clamped at the two crowns and the axle of the wheel. A light steerer rises between the two crowns and transmits steering action to the fork.
A single crown design places a lot of torque on the base of headtube join which limits the amount of travel that can be used in a standard 1.125 inch steerer (industry standard) to about 125mm. Any more and the rider risks breaking the front of their bike off or the legs off the crown when landing a big jump.
A double crown fork balances that torque out and very little stress is placed on the headtube which is why all long travel forks currently use this design.
A current industry initiative to create a 1.5 inch steerer standard will allow longer travel forks to be used in a single crown design but the standard is not yet well supported.
Other kinds of front fork
Cannondale make the Headshok which is a telescoping fork by class but instead of the legs moving, the steerer moves up and down inside the headtube. The design is patented by Cannondale and is not retrofittable.
Currently Whyte and Mercedes Benz are the only people who produce front linkage forks for their bikes. The MB bike is actually made by AMP Research who used to produce their own bikes with linkage forks but it now seems as if MB bought the design off AMP and rebranded it for their own means. K2 (formerly Proflex) used to make a linkage fork but have now reverted to using off the shelf telescoping units.
The fork achieves suspension action by driving some sort of parallelogram through the vertical portion of its path. Difficult to describe in words but in terms of suspension the fork is better left to those who know what they are buying or those who have too much money to care.
In engineering terms the design has more moving parts but can give a better suspension action. The design is proprietary to its manufacturer, rarely retrofittable and more expensive to manufacture. On the other hand it looks really cool.
Rear Suspension Overview
Rear suspension is provided by every major manufacturer in its bike lineup. Engineering evolution has ensured that over the past 10 years, three major types of rear suspension have made it through to the present. In any case most decent off road bikes will use either and air or spring shock unit from either Fox or Rock Shox. The range for higher quality bikes also extends to DT, Cane Creek and O2.
The demand for rear shock units meant that huge units intended for Moto X racing were rapidly miniaturized to fit within the needs of mountain bike suspension. During the evolution elastomer and MCU bumpers were rejected as too crappy.
The rear wheel rides upon a single large arm which is fixed to the front triangle by a single pivot. The arm also directly drives a shock unit directly. The amount of real wheel travel is dictated by the arrangement of the shock. But from 100mm (XC bikes) to 200mm (DH bikes) is fairly common. This design is great for wet and rough conditions as there is only one pivot to maintain but can have some issues with pedalling action locking out the suspension. This can be minimised but not eliminated with good design.
Orange make good examples of this method.
There are at least 4 distinct ways of building a linkage bike but they all share the same features being that the shock unit is not part of the frame, it is instead driven through one or more links attached to the the seat stays. The suspension can be very well tuned to affect the shock in different ways as the rear wheel runs through its travel.
The method is complex but allows for active suspension which means that the suspension is not affected by pedal or brake action. The extra metal needed to make this design can increase the weight but the quality advantages usually outweigh the extra weight. It is not quite as durable as a single pivot due to all the extra pivot points required.
Stiffness can be maintained by minimising the size of the rear triangle.
The Specialised FSR range has some good examples of this method.
This design is fairly uncommon these days but still appears in short travel XC racing bikes. The shock runs inline with the seat stays and is an integral part of the frame structure.
There is one pivot near the bottom bracket and another pivot near the rear axle on either the seat stay or the chain stay.
Rear wheel travel is equivalent to the amount of shock travel so the design is limited to about 50-75mm of travel at the rear axle. A very lightweight design but one which places a lot of lateral loading on the shock. There are durability issues with this method.
Some of the ones that didn't make it
Unified Rear Triangle.
The whole rear triangle and bottom bracket is a separate sub assembly. Pedal action doesn't affect the suspension in any way but as most variants locked out the suspension as soon as the rider got off the seat, it was effectively useless. Heavy too. This died out about 2 years ago.
A large beam of composite material rises away from the frame and flexes when vertical loads are placed upon it. This design is very stiff and durable as there are no pivots but it never really caught on. Probably due to the suspension being a bit untunable and the need to modify the riders pedaling technique. Softride still make road bikes with this system