Where do I need it?

Any outlet within six feet of a sink or tub/shower must be protected by a Ground Fault Circuit Interrupter. That means kitchen outlets, and bathroom outlets - including the one built into the medicine chest, and the outlets around the wet bar. In addition, outdoor outlets and those in the garage which are not "dedicated" must also be protected.
You can also put GFI outlets in other places where they will make you and your family safer. The kids´ bedroom and the game room jump to mind.
Also, see the categories Outdoor Lights, and Safety.

GFI

That funny outlet is also called a Ground Fault Circuit Interrupter. It is probably located somewhere close to water; the kitchen sink, the bathroom, outdoors, or the garage.
The outlet is designed to constantly measure the current in the black wire against the current in the white wire. A very small difference between the two wires will cause the outlet to shut off the current very quickly. If you drop the hair dryer into the bathtub, or run over the extension cord with the electric mower, the outlet shuts off.
The buttons are there so you can test the operation of the outlet. Since the outlet is close to water, and has metal moving parts, it is possible that the parts can rust. That is why it is important to test it once a month. Just push the TEST button. The button should pop out and the outlet should shut off. Then push the RESET button to turn the power back on.
If the outlet does not work like it should, you can replace it pretty inexpensively.
Also see the categories Safety, and When Do I Call An Electrician.

You can do it

First, make sure you have the correct GFI outlet. Look at the breaker which feeds the circuit you are working on. As a rule (but not guaranteed), kitchen circuits are 20 amps, and bathroom, garage, and outdoor outlets are 15 amps. The clerk at the hardware store will show you the correct one to buy.
Second, shut off the breaker feeding the circuit you are working on.
Remove the old outlet and spread the wires out so they don´t touch each other.
Turn the breaker back on and, with a circuit tester, determine which set (black and white) of wires is hot.
Shut the breaker back off.
Install the hot wires into the screw terminals marked LINE. The marking is on the back of the new outlet. It will also be marked to indicate the black terminal and the white terminal.
If you have more sets of wires, they will be installed in the terminals marked LOAD.
Screw the outlet into the box and turn the breaker back on. Push the reset button to "turn on" the outlet. Push the test button to make sure it is working properly, and then reset and enjoy.