browser settings
Microsoft’s Internet Explorer, Google Chrome and Safari should support .mp3-files natively.
Firefox should by default also enable .mp3-playback by using local ressources.
If this is not enabled, type “about:config” in the adress bar of firefox, search for “media.windows-media-foundation.enabled” (on Windows), or “media.gstreamer.enabled” (on Linux) and set it to “true”.

Linux & Firefox-specific
gstreamer should be available for all major Linux distributions and in most cases is already onboard by default; however, in some cases (e.g. on Arch Linux) the critical gstreamer-plugin for .mp3-support, “gst-libav“, needs to be installed separately (e.g. here for Arch). On Ubuntu, this plugin seems to be contained in “ubuntu-restricted-extras”; to install it (on Ubuntu), type “sudo apt-get install ubuntu-restricted-extras” in the Terminal.
If it still doens’t work, check that the basic codec plugins for gstreamer are installed as well (e.g. gst-plugins-good & gstreamer0.10-base & gstreamer0.10-ffmpeg & gstreamer0.10-good & lame; not sure all of them are needed)

In principle, the same should also apply for Firefox on Android.