Although it doesn't happen often, a good zap of static electricity can kill a PC, either while it's running or when you're or working on it. In this video, Patrick Norton and Michael Hand from DIY Tryin demonstrate how that can happen.
You've probably heard that you should ground yourself while working on computers, or wear an anti-static strap while you build that new rig you just spent several hundred dollars on. Truth be told, most of us don't bother. However, it's important to go into the risk with both eyes open, and using a Van de Graaff generator as the source for their sparks, Patrick and Michael proceed to zap their PC's critical components to see what lives and what dies.
The end result is that sadly just about everything dies, most of it pretty quickly. Obviously if you're in a low-humidity, static-free environment (like a PC building lab or a data centre) you have less to worry about, but if you're assembling your new PC on a carpeted floor, you may want to at least try to remember to ground yourself on the metal of the case every now and again if you don't feel like wearing a strap.
If your home is particularly prone to static electricity, we have some tips to keep your PC safe.
Can Static Electricity Kill Your PC? [DIY Tryin]