Basically all you need is a folder on your server that contains the audio player assets - "player.swf" and "audio-player.js" - and your MP3 files.
Create a new folder on your desktop and name it something like AudioPlayer. Copy and paste or drag and drop your MP3 file(s) into this.
Go to ...
http://www.1pixelout.net/code/audio-player-wordpress-plugin/
and download the Audio-Player.zip
Open the folder, copy "player.swf" and "audio-player.js" and paste them into the folder with the MP3s.
Using your FTP application, upload this folder to your server and place it in the main folder that contains all your website folders and files.
If you want a really simple way to do this, download the freeware Easy iWeb Publisher....
http://www.plyxim.com/free/
Publish your website to a folder on your desktop, drag your AudioPlayer folder into this and then drag the website folder onto Easy iWeb Publisher.
The URLs that you use in the code are derived from this AudioPlayer folder.....
http://www.mydomainname/AudioPlayer/player.swf
http://www.mydomainname/AudioPlayer/NameOfSong.mp3
....etc.