The term Podcasting or Podcast as it is also known, plays upon the terms broadcasting and webcasting and is derived from the name of the iPod portable music player, the playback device of choice of many early podcast listeners. Podcasts are not directly associated with Apple's iPod device or iTunes jukebox software. Podcasting is similar to time-shifted video software and devices like TiVo, which let you watch what you want when you want by recording and storing video, except that podcasting is used for audio and is currently free of charge. Note, however, that this technology can be used to pull any kind of file, including software updates, pictures, and videos.
In short, Podcasting involves the recording of internet radio or similar internet audio programs. These recordings are then made available for download to portable digital audio device. You can listen to the Podcast internet radio program while you are away from your computer or at a different time than the original program was broadcast.
The term Podcasting was invented by Dannie J. Gregoire, who registered the domain name podcasting.com, discovered and reported on by Dave Slusher of the evil genius chronicles and made popular by former MTV VJ and Dutch weblogger Adam Curry's original ipodder.org ipodder script. RSS pioneer Dave Winer describes succinctly the technology used to pull digital audio (e.g., especially MP3) files from websites down to computers and devices where the audio can be played back at a listener's convenience. The recovery of MP3 links from distributed weblogs and distribution of the aggregated list using RSS had been demonstrated by Stephen Downes's Ed Radio, launched June 9, 2004. Podcasting was developed, according to Curry in August, 2004.
The ipodder.org definition written by Dave Winer: "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. A Podcast works in 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."
Adding to a number of ad hoc, proto-podcasting techniques for automatically downloading audio files, podcasting proper became popular through association with blogs (in particular MP3 blogs), the XML-based file format called RSS (Really Simple Syndication), and the polling applications called feed readers or news readers that poll and download RSS files. Blogs, often being self-published websites, provided a convenient means for individuals to self-publish audio files online. RSS already gave websites and blogs a means to summarize or list new content added to the site. Individuals already used RSS to poll websites for new content. Thus, the addition of audio file listings to RSS, and the addition of audio file downloading to RSS feed readers built upon the feed reader's existing methods for polling and downloading files, and upon the existing "reader driven" interaction with content publishers.
Podcasts are different from broadcasts and webcasts in that it "casts" audio not by a mechanism of centrally pushing audio out to listeners, but by the mechanism of the (distributed) listeners pulling (downloading) the audio files automatically. Podcasters publish (or "podcast") audio files, even in the likeness of radio shows, but it is the individual listener who initiates the "cast" through their subscription and automatic download of the audio program.
A Podcast also also utilizes the combination of audio files and RSS to publish descriptive data and metadata associated with the audio. RSS files used in a podcast include dates, titles, descriptions, and links to audio files. These links to audio files within RSS, which are used by podcasting applications, are defined by the enclosure element in RSS 2.0, and all of the original podcasting proper applications were built to work with RSS 2.0.
[update]
In June 2005 Apple announced it was supporting Podcasting in iTunes 4.9 through its music store. Downloading and subscribing to Podcasts from the store has been made a simple and straight forward process as only Apple can do.
Source: Wikipedia, the free encyclopedia.

