A person could literally write a book on the explanation of shock basics, types, and options, and some have. So I will just briefly touch on the types you have specifically asked about. There are many more shock types and options out there. For the typical daily driver and weekend off-roader this information is meaningless as they have no use for anything more than your run-of-the-mill shock. If you guys truly would like to know more about the similarities and differences, pros and cons, uses, etc. then I suggest you spend some quality time doing some plain old research. Here’s an excellent article to get you started beyond what I have explained below. Here: Shock Bible - Part 1
It is enough to make most people’s head explode.
Hydraulic (hydro) and gas charged (nitro) shocks have the same amount of oil; the difference is the gas charge.
A hydraulic shock has equal amounts of dampening force in both directions. Hydraulic shocks have a tendency to fade as the oil heats up and gets agitated by the shock, creating air bubbles. These air bubbles cause the shocks dampening ability to fade.
Gas charged shocks usually are pressurized with Nitrogen. The nitrogen charge helps keep the air bubbles under control, so they will not affect the shocks performance. The combination of the inert gas and the higher pressure makes the hydraulic fluid more resistant to foaming and boiling, allowing the shock to operate over a wider range of temperatures. A gas shock will extend on its own due to the gas charge.
General rule of thumb is that hydraulic shocks ride better; nitrogen charged shocks are stiffer, but last longer and don't have as many shock fade problems.
A coil-over is designed to package the shock and spring(s) as a single system, which allows us to design for a given frequency (ride and handling), and then easily tune to achieve a desired suspension height. Coil-over’s are completely rebuildable and revalveable, and are known for their tuneability and adjustability.
Emulsion shocks are a gas-charged mono-tube shock. In this case, the shock has no reservoir or floating piston. The nitrogen charge is contained in the pressure tube along with the oil in an “emulsion”. Emulsion shocks are best suited to light weight and/or low-speed applications where the shocks will not be subject to excessive heat. Emulsion shocks are more compact and more economical than a remote reservoir. An emulsion shock can only be mounted right side up, but are usually adequate for moderate rock crawling.
Remote reservoirs are a nitrogen-charged mono-tube shock as well. However, a remote reservoir uses a “reservoir” to house a nitrogen charge and floating piston, allowing the use of a shorter tube than would otherwise be necessary. The reservoir is connected to the pressure tube with a short, flexible hydraulic hose. Depending on the position of the piston in its travel, there will also be a certain amount of oil in the remote reservoir. Remote reservoirs are designed to increase the oil capacity. This increase in oil capacity helps prevent the shock oil from overheating and breaking down. When shock oil overheats the damping effect of the shock is lessened. Remote reservoirs are typically used in high-speed applications, or where the vehicle will be subject to frequent extreme off camber situations. A remote reservoir can be mounted at any angle.