You've got (or your learners have got) an MP3 player…? Then you've got just about all it takes to be able to podcast…
Podcasting — a portmanteau of the terms "iPod" and "broadcasting" — involves publishing MP3 (audio) files on the Internet, downloading and listening to them.
It has become hugely popular [statistics from the BBC] and is one of the technologies we might consider using with language learners (though, says The Guardian, which also claims to have invented the word, it is "…still a format awaiting a purpose".
Where to begin
This short concise article on teachingenglish.org.uk is probably the easiest place to begin to form an overview of what podcasting involves and what you need to be able to do it.
What use could you make of podcasts with your learners? Techlearning.com explains How Educators Can Use This New Technology.
Where to find good podcasts
On Yahoo, as on other search engines, you can search for podcasts on a huge variety of topics.
You also have specialised podcasting portals like Podcastalley.com, where you can also search under different genres (including, eg., education) as well as seeing what's most popular.
The mainstream online media — the New York Times, for example — has also quickly picked up on the popularity of podcasts.
For more "general interest" podcasts, LifeHacker.com has an excellent article on where to find them.
Podcasts for English teachers and learners
For podcasts specifically for ELT, there are several places you could try:
On the "Podcasting ELT" Yahoo Group (join it, if you are interested in podcasting), you will find lots more links relating to podcasting.