Computer parts are less delicate then most people think. However, there are some rules that everyone should know before proceeding with a repair.
Unplug the computer
Of course you would say, the PC must be unplugged before all. That can avoid short-circuits while working with electronic parts like such as the motherboard, for instance.
Static is electricity. This kind of thing that makes you wince when you touch your car door is about thousands of time higher than the maximum charge that a computer part can receive. Make sure to always wear cloathes made of non-synthetic materials and to touch metal before touching a part. You might also want to wear latex gloves.
Predict bad moves
Observe before acting and be patient. Take a look at all of your interventions before applying them. If your screewdriver suddenly slip and hit another part, it might make your day a bit darker.
Make some space
Repairs often take more space then expected. It is easier to work and to prevent bad situations when we always make sure to have some space left to leave another part or tool.