F1 Dampers
F1 cars generally use concentric linear (i.e. telescopic) dampers, they use valves on the damper rod to vary the damping rate both in bump\rebound at both high and low dampers speeds. Williams Front damper shown below



NOTE: High and load damper speeds does not relate to the cars speed, but the rate the damper rod moves through the body, i.e. low speed could be from roll or pitch through braking\cornering, high speed is more likely from road imperfections, hence separating the two allows the chassis engineer to create an element of roll\pitch stiffness from the dampers alone (i.e. not the roll bars). Williams Front damper shown below



Adjustment is often via screw adjusters on the top of the damper body, generally the damping range is set into the valving before the race weekend and once a satisfactory set up (i.e. pre race or Q) is achieved the teams remove the adjustable dampers and build a non adjustable dampers valved to match the final settings on the adjustable unit.



Through rod Damper
Sachs have now developed a through rod damper which has the damper rod passing directly through the damper body, this negates the need for the nitrogen charged chamber at the top of the damper to displace the internal volume changes of the damper rod passing up and down inside the damper body. This removes the initial spring effect of the nitrogen cylinder (known as Nose friction) making the damper more sensitive.

SACHs Through Rod Damper PDF

Through rod dampers also have one other benefit in that they allow a range of installation options. Assuming the damper does not use coil over springs (i.e. torsion bars instead) the damper can be mounted both in compression (in bump) as with conventional dampers, when the two operating ends of the suspension linkage are moving closer. But also in tension (in bump) when the two operating ends of the suspension linkage are moving apart.

This allows the designer to place a damper in different positions to the norm. Toyota used a Through Rod Damper in tension on the TF102 (2002) car. Placing the dampers (red) to the side of the gearbox casing and when in bump the high pushrod operated linkage (yellow) moved away from the lower damper (static) mount, effectively operating the damper in reverse and clearing the top of the gearbox of obstructions for aerodynamic benefits.



Radial damper
Sachs have now also produced a damper for Ferrari, that uses a radial motion (like a lever arm damper son old British cars) this is more compact and fits in better with the motion of the pushrod linkages. Theres no other info on this damper but I suspect the design also creates a fixed internal volume regardless of damper rod position.

A few notes on dampers
One is there lack of sensitivity to heat, as the dampers use an oil based fluid to provide the damping effect, the change in the fluid viscosity from heat should also alter the damping rates. Developments in fluids and heat reactive valving have prevented heat related changes in damping. Often the rear dampers are set inside a niche in the gearbox (i.e McLaren below), placing them in a hot working environment, this is not an issue for the teams or damper suppliers.



Also the use of coil over springs does not affect the dampers efficiency, normally a coil over spring will induce a bending moment into the damper placing side loads into the bushes creating undue friction, the side loads are still their but bushing materials have improved to negate this problem. Jordan and Williams (Jordan below) used coil springs over torsion bars in 2003.