Heres what you do...
If you are not comfortable with playing with wiring while hot, like everyone else said, turn the circuit off. If you're still not comfortable, call an electrician to do it. Shouldnt take more than 3 minutes.
1. Remove plate
2. Carefully pull out recepticle. (whether its hot or not)
3. That is what we call "backstabbing idiots" when they stab the wires into the recepticle.
Now, you have 3 options for removing the wires...
For all backstabs, you dont have to remove the screws, those do not hold the wires on backstabbed recepticles.
A. Cut the wires with wirecutters, and restrip them.
B. Insert a toothpick, or other similar item into a slot near where the wires are going in. This will release the wire so it can be pulled out.
C. Grab the wires, and twist and pull... It'll come out eventually.
D. If you cut the wires, strip the wire. There is a strip guage on the recepticle.
4. When removing the wires... Think of this, the hot wire (black) is the first to break, last to make.
5. Remove the hot (black) wires from the recepticle
6. Remove the neutral (white) wires from the recepticle
7. Remove the ground wire.
8. Discard old recepticle, and grab the new one.
9. Reconnect the ground wire
10. Reconnect the neutral (white) wires.
* You can either do the backstab thing like the original, or bend the wire around the screw terminals. **
11. Reconnect the hot (black) wires.
12. Secure the recepticle to the box with the supplied screws.
13. Reinstall cover, making sure not to overtigten the screw.
14. Reapply power (if you turned off the circut), and check for power.
BTW... Yes, this can be safely done with the power ON, I do it all the time.
If this was a GFI recepticle, you would have to make sure you connect the wires the same way by looking at which pair is the "line" and "load". Improperly connecting these will result in a non working plug, and everything behind the GFI. If it is wired properly and by the looks of it, there is probably another GFI above in the circut, or there is a GFI breaker installed.
How do you check if the circuit is GFI protected?
Touch the neutral (white) wires and ground (green or bare) wires from the "line" side of the circuit together. If the circuit doesnt shut off, its not GFI protected and should be corrected by installing a GFI or consulting an electrician.
This is pretty much a walkthrough on how it can be done. Remember, electricity is dangerous, and if you are not comfortable with this, call an electrician