Thermal paste is used to create a solid contact between a computers Processor and the Processor's cooling unit. Or to put it simpler, without it your CPU is going to get hot enough to cook steak and burn out. The hard part with this is getting a nice even coat. You dont want it to be spotty.
An Anti-static mat, or wristband if you prefer that option, is used to keep you grounded and completely free of static electricity. Much like when you touch a doorknob or other metal item after walking over a carpet, a static shock can transfer to your components and damage or fry them. Unfortunately for us as well, (Don't quote me on the exact number here) it only takes a static discharge 1/16 of what we can physically feel (As in 16/16 is the lowest static shock you can feel) to cause this damage or destruction. In short, always wear one when working with delicate components (CPU CPU CPU, and ram). It's not as likely with hard items like Graphics cards, but still a good habit.