Download and install Audacity. (See References)
Find the song you want to edit and open Audacity. Go to "Project" and "Import" on the top menu bar and browse for the song you are editing.
Listen to the song once or twice to figure out what you want to do to edit it.
Select a part of the song. Use the tool that looks like the line in word-processing documents and click the track and then drag to select the portion you want. Double check that the part you selected is what you wanted by clicking the green "Play" button to play it. If you need to adjust the selection, hold the cursor over a part of it until it becomes a finger and then click and drag.
Use the cut and paste tools like you would with text: Select the part you want and cut it; then click where you want it to go and paste it. You can find these commands in the "Edit" drop-down menu.
Fade in and out are best used at the beginning and end of a song to make it stop and start a little more natural. You can use these tools to put together a medley of tunes, as well.
Use the "Silence" tool to replace a section with silence rather than deleting it. The "Join" button can be used to put more than one song clip piece together into one.
Export the finished product by going to "File" and choosing "Export MP3."