Heat sink paste is a type of thermal compound used to fill in the small gaps between the CPU and the heat sink to improve the transfer of heat. Most CPUs require a heat sink for proper cooling, but even the best ones will not work properly if there are microscopic imperfections in contact. These tiny imperfections, which inevitably contain air (a poor conductor of heat) prevent the transfer of heat from the CPU to the metal heat sink. The purpose of Thermal Paste is to bridge the gap between these two surfaces, allowing heat to pass through freely (see image above).
The most effective (and expensive) thermal formulations consist almost entirely of liquid metal, usually a variation of the alloy Galinstan. These can attain thermal conductivities well above 13 W/m*K, but are difficult to apply evenly and have a high risk of malfunction due to spillage (gallium is highly corrosive to aluminum).
Ceramic and silicon-based formulations don’t have the same electric conducting properties as metals, and are typically much more affordable and easier to use than metallic pastes. They are also compatible with many types of electrical systems and can withstand high operating temperatures.