A "ground" is, generally speaking, where you want the electricity to go. Otherwise (and I'm grossly oversimplifying) it basically goes "everywhere".
If you wear a grounding strap, virtually all of the electricity goes into YOU. Which, when you're working with the extremely low amounts of electricity generated by static whilst you're working on a computer, is less harmful to you than it is to the components you're messing with (you literally won't even feel it most of the time, it can fry unprotected electronics).
However, in a situation where you're working with more power (something plugged into a wall, or even the unshielded capacitors inside a computer's power supply - which is why you're NEVER supposed to open them up), all that electricity going into YOU is a major problem - one that usually leads to a charbroiled technician.
So in a nutshell:
DO use grounding straps in situations where the voltages/amps are very low and dangers of shock are more concern to the machine than you. Know that you WILL take all the shocks.
DON'T use a grounding strap in situations where your body can't take the potential shock.
When in doubt, don't use a strap. Accidentally frying a component inside the computer isn't nearly as bad (or as expensive) as accidentally frying the technician.
The last thing you want if you accidentally touch a hot chassis is for your body to be grounded. – Fiasco Labs – 2012-08-05T07:13:20.690
Also, to the OP, you should unplug your devices when you're working on them unless it's like a hotswap drive or something. – cutrightjm – 2012-08-05T07:18:35.493
@ekaj Yep, when you become the ground circuit and the current path through your chest causes fibrillation, you become a statistic. Or did you not know that? – Fiasco Labs – 2012-08-05T07:24:02.410
If you're worried about getting shocked, you actually want to do the opposite and insulate yourself from any potential grounds. The idea is to become the path of most resistance for any current to take. DO NOT GROUND YOURSELF if you will potentially be coming in contact with live electronics. That's a good way to suffer shocks, electrical burns, and heart failure. – Joel Cornett – 2012-08-05T14:09:57.613
1@Joel Cornett - It's why all the electricians I know favor fiberglass ladders. There are times when you will be working on live circuits (ballast changeouts in production plants for example), plus the healthy assumption that all circuits are live until proved otherwise. – Fiasco Labs – 2012-08-05T15:32:35.297
@FiascoLabs Yeah, I knew that. It's common sense. Hence the "unplug your computer" part. – cutrightjm – 2012-08-05T17:21:25.493
@ekaj - It's not so common knowledge, I've heard a lot of oddball explanations out there that could get you killed. A lot of people don't understand that ground is a state of equal potiential, therefore with the computer power unplugged and the case disconnected from earth ground, the purpose of the wrist strap is to connect your body to the computer's ground plane so you are at equal potential to the computer ground so no arken'sparks occur. Same goes for earth ground in a power mains circuit. Chassis are grounded so exposed metal will be at ground potential even if there's an internal short. – Fiasco Labs – 2012-08-05T17:33:27.123