HydraulicAll shocks are hydraulic in some way, they all have fluid inside them. But a pure Hydraulic shock or strut would just have fluid and NO nitrogen gas inside. Most shocks and struts made from the 1930's up to probably the 70's were of this design. Without Nitrogen gas inside, their dampening ability was extremely limited. Cavitation/aeration was a common problem, and any dampening ability the unit did have fades very quickly. This design is old, and the Companies that make our products today have pretty much stopped making these kinds of units anymore.
Twin-tube Low Pressure GasA shock absorber or strut can be a low pressure gas, twin tube design. This basic design has not changed too much since the 60's, but there have been some refinements over the years. This design has some characteristics, such as a soft valving, and usually costs less to manufacture than a Monotube or Adjustable type shock absorber. They are usually pressurized with some Nitrogen gas, usually 100 psi or less. In this design the hydraulic oil and gas are mixed together in the same chamber, which, while dampening, causes some aeration/foaming. Monroe Auto Equipment, Bilstein, and KYB Excel-G still make units like these.
Mono-tube High Pressure GasA shock absorber or strut can be a high pressure, mono tube design. These are a more modern type design, and have characteristics of their own. The monotube design allows them to operate cooler. Monotubes are under high pressure, from around 200 psi, to as high as 360 psi. The hydraulic oil and Nitrogen gas are in separate chambers, separated by a floating piston. This allows the shock/strut to function without any aeration or foaming. Monotube shocks usually have a stiff valving, and traditionally cost more to manufacture. They are also able to withstand more punishment, and offer higher dampening ability. (for instance, most racing shocks, from Nascar to Formula One, are of a monotube design). Great care and engineering is taken into consideration when manufacturing a monotube, and many have a very specific valving. This is years of racing technology adapted for street and off-road use. Some of the companies that make monotube shocks are Bilstein , and KYB.
Adjustable UnitsDepending on the manufacturer and the vehicle, an aftermarket adjustable shock absorber can be a pure hydraulic unit, a low pressure gas twin tube, or high pressure gas monotube unit. Most units are adjustable in that you can adjust their valving, to cause the unit to become softer or stiffer. This allows you to have better control in setting up the suspension for your desired driving/handling. Another trick is to start on the lowest setting, drive on that for a year or so, then adjust the units to be a little firmer, which brings the suspension back up to being like new, without purchasing new units. You still have to be the type of person who enjoys crawling under your vehicle to adjust them in most cases. Which is not a big deal for a race car, but can be cumbersome for a daily driver. Some companies, like KYB, make externally adjustable units such as their AGX struts and shocks.
Coil-OversOn a car with struts, a coil-over strut allows you to adjust the HEIGHT of the vehicle. It incorporates a threaded sleeve that fits over the strut, with spring perches you can move up and down. In most cases, these units also require special springs. On a car with shocks, a coil-over shock will do the same thing, allow you to change the height of the vehicle some, and in some cases, add some support when you have eliminated the factory leaf or coil springs. The down side to coil-over struts is they are a pure racing suspension setup, requiring setup time and adjustments. On a daily driver, it can be rough riding and high maintenance. If you do not own a set of scales to properly set the car up for a particular track or use, their use can be very limited in the real world. Be sure your shop or mechanic who installs coil overs knows how to set them up properly.