Assuming you're using Ubuntu 7.04 or later, the easiest way to install MP3 playback in Ubuntu is to just try to play an MP3 file. So start off by finding an MP3 file and then double-click it.

If you're using Ubuntu 6.10 or earlier, the community documentation will show you other methods for installing MP3 playback.

Once you do that, Totem will try to play the MP3 file and then realize you do not have the necessary codecs to play MP3 files. It'll ask if you want to search for the appropriate codec. Click onYes to continue.

You'll then see a list of possible codecs to install. As you can see, the first one shown enables MP3 playback. So check (or tick) the box next to Gstreamer extra plugins.

If you haven't already enabled extra repositories, you'll be asked if you want to install community-maintained software. This will be necessary in order to install the codecs. If you don't know what a repository is, read this overview of software installation in Ubuntu.

Wait for the package information to download. Ubuntu is checking to see what software packages and versions are currently available in the community-maintained software repositories.

Once that finishes, try once again to check (or tick) the box next to Gstreamer extra plugins.

You'll be warned that this is restricted software and asked if you want to continue. I think this is a generic warning for all non-free software (by the way, non-free refers to freedom, not money). As far as I know, MP3 codecs are legal for users to install in all countries. If you have qualms about installing the necessary codecs, read more about MP3 at Wikipedia. Otherwise, click OK to proceed.