Basically, podcasts are free broadcasts that you can download as free audio files and listen to on your iPod (or on your computer with iTunes).
Because it's really easy for people to record (publish) themselves and submit the recording to a site where there are thousands of podcasts, it's become sort of popular. Some churches are using it to "broadcast" sermons, other uses are radio programs and such. For example, if you miss CarTalk on NRP one Saturday- well, you can go look to see if it's available in Podcast form and download it for free.
Here are some links that you might find interesting: