You have an MP3 file (a podcast, a song of your band, a lecture) and you want to share it with the visitors of your site. You can just put a link to the MP3 and let people download it or you can include it as an object and let the browser play it with a plug-in like QuickTime (and most of the times also crash the browser). Most people want to preview the audio and not wait until it downloads completely. You can play a partial MP3 file in audio player, but many people aren't aware of that.
But there's another away: a Flash player that allows you to pause and rewind the MP3. It's like an audio version of Google Video or YouTube, with the difference that the file is hosted on your server (or on another server). While there are many solutions based on Flash, this is the one I like the most. It's a modified version of Odeo player, that's used to play podcasts on odeo.com:
You just have to replace [MP3 file address] with the actual address.
Here you can see how it looks (in this audio, Marissa Mayer talks about innovation at Google):
Update (April 28th, 2010): Odeo.com is down and the player no longer works. "Odeo will be down for maintenance and site improvements. We will restore service as soon as possible. Sorry for the inconvenience," says the site. Another player you can try until odeo.com is back online is Google Reader's player.
6 hours ago