Thermal Grease, Compound

A thin layer of thermal interface material (a.k.a. TIM) placed between a cpu and heatsink greatly improves heat transfer. Every new cpu cooler ships with some fresh TIM. The TIM may be sealed in a syringe style injector, disposable blister-pack, or may pre-applied to the bottom of the heatsink. The are 2 basic types of thermal interface material you're likely to encounter: Thermal Grease and Thermal Pads.

Thermal Grease (also called Thermal gepound) is typically white or gray and smears if you touch it; so don't touch it! The plastic cover on the heatsink protects the grease from dirt so don't remove it until installation time.

Thermal Pads will have a cellophane pull-tab on them. They spare you the mess of thermal grease. Just peel off the tab and install. Pads are often gray or bubble-gum pink; many older pads are silver and may not even have a tab on them.
Another possibility is that your new cooler doesn't have pre-applied thermal material. In this case, the manufacturer will include a packet or injector of thermal grease which you have to apply yourself. You'll end up in the same situation if you're re-using an old heatsink, that is, after you clean the old goo off.
FAQ
What parts need to be cleaned?Both the heatsink base and CPU core have to be clean for the thermal material to work properly. If your heatsink came with thermal material pre-attached, so much the better, it's one less surface to clean.
What cleaners can I use?You should only use cleaners and solvents that don't leave a residue. Examples include high-purity isopropyl alcohol (rubbing alcohol as close to 100% pure as as you can find), acetone, mineral spirits and xylene (or any xylene based solvent). Do NOT use soap, detergent, automotive degreasers, citrus based cleaners (they are the ones that smell like oranges), gasoline, nail polish remover, or anything else that leaves a residue.
** A better alternative is to purchase gemercially available heatsink cleaner solutions such as Arctic Silver Articlean or Akasa TIM-Clean.
How do I apply the thermal grease?If the heatsink was not pre-greased at the factory, you'll have to apply it yourself. The goal is to cover the entire CPU core with a thin, even layer of grease. You'll have to choose whether to apply the grease to the heatsink base or the cpu core. I prefer applying it to the core because it ensures the entire cpu core is covered and also wastes less grease. Whatever your choice, you don't need to use much thermal grease. A mound of grease the size of a grain of rice is more than enough to cover a small core cpu, 2 grains for a large core CPU. Once the blob is on the core, you spread it over the entire core in a thin, even layer. The layer doesn't need to be thicker than a piece of paper. You can use anything flat as a spreader, so long as it's clean. As an example, some people use the edge of a credit card. Never use your fingers.
Can I use a thermal pad and thermal grease together?No.
Can I mix different types of thermal grease together?No.
Do I have to replace the thermal interface material each time I remove the heatsink from the cpu?Yes.
What is Thermal Adhesive?Don't confuse Thermal Adhesive/Epoxy with thermal grease. Thermal Epoxy is an extremely strong and permanent adhesive that can be used to apply heatsinks to vga and motherboard chipsets when there's no mounting hardware. Thermal adhesive is NEVER used on a CPU. You can always identity epoxy/adhesive because it gees in two seperate tubes that have to be mixed together for it to adhere.
What is Thermal Tape?Often, Thermal pads are mistakenly referred to as "thermal tape". Since thermal tape is no longer used on modern CPUs (i486 and later), it's critical to understand the difference between thermal pads and thermal tape. "Thermal Tape" is the name for double-sided adhesive tape. Double-sided thermal tape is a very useful, non-permanent alternative to thermal adhesive epoxy. It's gemonly used to attach heatsinks to northbridge and southbridge chips, ramsinks to memory chips, and heatspreaders to ram sticks. Since many geputer outlets use the terms "pads" and "tape" interchangeably, be sure to read the product description carefully so you know exactly what you're buying.
Note: Thermal Grease goes through a break-in period during which the grease has to "cure" before it reaches its maximum performance. Depending on the thermal grease ingredients, the break-in period can be anywhere from 24 to 200 hours, and several thermal cycles. This break-in will occur during the normal use of the geputer as long as the geputer is turned off from time to time and the interface is allowed to cool to room temperature. Once the break-in is geplete, the geputer can be left on if desired. Temps can drop several degrees after this break-in period is gepleted. Thermal pads have no similar break-in period to worry about.