Being picky, this belongs in CPU/Overclocking, not GH.
Peltier's are more accurately called TEC - Thermoelectric coolers - although most people call them Peltiers anyway. Thermoelectric coolers are solid state heat pumps that operate on the Peltier effect, the theory that there is a heating or cooling effect when electric current passes through two conductors. A voltage applied to the free ends of two dissimilar materials creates a temperature difference. With this temperature difference, Peltier cooling will cause heat to move from one end to the other. A typical thermoelectric cooler will consist of an array of p- and n- type semiconductor elements that act as the two dissimilar conductors. The array of elements is soldered between two ceramic plates, electrically in series and thermally in parallel. As a dc current passes through one or more pairs of elements from n- to p-, there is a decrease in temperature at the junction ("cold side") resulting in the absorption of heat from the environment. The heat is carried through the cooler by electron transport and released on the opposite ("hot") side as the electrons move from a high to low energy state. The heat pumping capacity of a cooler is proportional to the current and the number of pairs of n- and p- type elements (or couples). (above taken from
here).
In plain English, a Peltier is an inefficient refrigerator. You pump in a lot of current and this creates a cold plate that cools the CPU while creating a lot more "waste" heat on the "hot" side. So if you have a 70W CPU that runs at about 60C under load and you put in a TEC you could see temperatures in the CPU drop to 20-30C, but your heat output of the CPU, Peltier combo may be 200W (or probably more). So you cool your CPU, but you heat up everything else.
Neoprene is used to minimize condensation which can short out components.
You would use these things to lower the core temperature of your CPU or other semiconductor component. Probably with the goal of increasing the efficiency of the CMOS transistors inside thus allowing you to increase the core clock speed beyond specification.
I have used TEC's extensively at work and at home and they are very touchy things. You need to be careful of many things (Power supply overloading, moisture, overheating the RAM/chipset/graphics/etc.) and you don't get that much for your effort. I don't recommend playing with them unless you are willing to gamble with your computer components. Ideally Peltiers make a lot more sense in combination with pulsed power regulatoring thermostats and heat pumps or water coolers. These avoid most of the problems associated with Peltiers: condensation is avoided by setting the thermostat to a value above the dew point and they can also stop power supply overload, and a heat pump/watercooler removes the waste heat from inside the case).
My TEC that I have used for hobby experiments has a pulsed power module that reads the temperature of load device , and adjusts the power input to the TEC to maintain a constant temperature. I built it almost entirely just to see if I could - additionally TEC's and the Peltier effect are pretty interesting by themselves. But I have not actually used my TEC to cool my main computer's CPU. My data and computer are worth too much to me to gamble with them.