What is Phonon?
KDE versions 2 and 3 uses aRts as their multimedia framework. Over the lifespan of these releases aRts fell behind newer frameworks such as Xine and GStreamer and is by now unmaintained. Learning from these mistakes, KDE implemented Phonon for KDE 4.
Phonon was created to allow KDE 4 to be independent of any single multimedia framework such as GStreamer or Xine and to provide a stable API for KDE 4's lifetime. As an added bonus, it simplifies development by providing a simple API that can work across different platforms such as Windwows and OS X.
The result of all this for you, the user, is that you get to choose the multimedia framework that works best for you. For Linux users the currently recommended choice is Xine, but this may or may not hold true for you.
Another result is that playback problems in Amarok are typically not Amarok issues, but rather Phonon issues. It is for that reason that this page exist.
The Phonon settings can be found at: Settings --> Configure Amarok --> Playback --> Sound System Configuration.
Selecting an output device
Depending on your OS and distribution, you may need to use ALSA, OSS, PulseAudio, or something else as your default playback device. Find one that works for you with the "Test" button and apply to all types of output. If you would like different types of output to go with different devices, this is certainly possible as well.
Selecting a multimedia backend
Depending on your OS you may see options such as Xine, GStreamer, VLC, or DS9 (Windows only) here. The recommended one for Linux users is Xine. VLC is recommended for Windows users, but is experimental like the rest of KDE for Windows.
If you don't see Xine or the particular back-end you want here you may need to install the package for it. For most distributions the package for Xine, for example, would be phonon-backend-xine. If you need more detailed instructions you should see your distro's section on the download page.
For information on playing MP3's see the MP3 page.