Podcasting is a way of publishing sound files to a website  that allows users to subscribe to the site and receive new files as they are posted. Most podcasts are downloaded automatically by subscribers, podcasting allows individuals to have a self-published, syndicated radio show. Users subscribe to podcasts using podcast reader software, which periodically checks for and downloads new content. However, podcasting does not require an iPod: any digital audio player or computer with the appropriate software can play podcasts. Podcasting can be thought of as an audio magazine subscription, in that a subscriber receives programs without having to get them, and can listen to them at leisure. It can also be described as the internet equivalent of timeshift-capable digital video recorders (DVRs) such as TiVo, which let users automatically record and store television programs for later viewing.  (wikipedia)

Think how a desktop aggregator works. You subscribe to a set of feeds, and then can easily view the new stuff from all of the feeds together, or each feed separately. Podcasting works the same way, with one exception. Instead of reading the new content on a computer screen, you listen to the new content on an iPod or iPod-like device. Think of your iPod as having a set of subscriptions that are checked regularly for updates. Today there are a limited number of programs available this way. The format used is RSS 2.0 with enclosures(ipodder.org)

Podcasting was developed in part thanks to Adam Curry's original iPodder script and the success it fostered since August, 2004. Dave Winer had added an "enclosure" element to the RSS 2.0 specification, Winer's company, added the enclosure feature to its blogging software and built-in aggregators.    (wikipedia)