What the heck is Podcasting, I hear you ask... or if you ain't asking, I'll tell you anyways :)
Podcasting is the act of producing Podcasts.
Recently (August 2004 ?), several people (including Adam Curry and Dave Winer - now acknowledged as two of the 'pioneers' of Podcasting) formed a small community who produced short audio commentaries, as an adjunct to their blogs (Web Logs -> 'b'logs -> blogs).
These commentaries are integrated as mp3 audio file 'enclosures' into an RSS (Real Simple Syndication) 2.0 'feed' associated with a blog.
This means that essentially anyone with a computer connected to the Internet can 'subscribe' (via a special-purpose program such as iPodder, iPodderX or jPodder) to a number of such feeds, and have these short audio 'podcasts' automatically fetched (say, overnight) into their MP3 players (e.g. an Apple iPod - hence the name Podcast).
Having done this, the recipient is free to listen to (or skip over) the short programs at their leisure, using the MP3 player - while walking, cycling, driving, on airplanes, wherever the fancy takes them.
More recently, interest in this method of producing and distributing audio programs has exploded, and now major players in the broadcasting industry are sitting up and taking notice. The BBC (British Broadcasting Corporation), long known for high-quality and well-researched news and documentaries, now themselves produce at least one (no doubt many more to come) podcast feed - 'In Our Time' with Melvyn Bragg (of The South Bank Show arts review fame).
At the other extreme of the scale, it's relatively simple now for anyone with a computer, sound card and microphone to produce their own Podcasts - and many people are doing just that. Consequently some are saying that this will lead to a democratic revolution in broadcasting - no longer will broadcasting be the domain of the media barons such as Rupert Murdoch, Kerry Packer and their ilk.
What I would recommend you do is to download and install one of the players available at http://www.ipodder.com or http://www.jpodder.org - then subscribe to a few feeds (there are preloaded lists in the 'Podder programs, and you can add more feeds as you discover them), download some podcasts and listen to them in the MP3 player in your computer - just to get a sampling of what's out there. Come on in, the Podcasting's fine!!
A warning though: there's no censorship (yet!), and some of the podcasts are more- or less-liberally peppered with profanities - so you'll want to vet these to your own tastes and standards before, say, you let your children start listening regularly. There's also a spread of political view; most of the ones I've heard so far tend to be rather leftish.... Your Mileage May Vary!!
Some of my favourites (you should be able to find these readily in the preloaded lists) are:
In Our Time - The above-mentioned, very eclectic discussion panel programme by the BBC.
The G'Day World Podcast - tech-related stuff, Aussie Slang lessons, and all manner of other tangents.
The Engadget Podcast - more techie stuff :) - about PDA's, smart phones, media players, TiVo's etc. in particular.
The Firesign Podcast - oldish American comedy from the 60's - mostly pretty innocuous from what I've heard so far.
Adam Curry's Daily Source Code - One of the pioneers - profanity alert! - Serves as a 'hub' with references to a lot of what is happening in the Podcasting world.
Career Opportunities - some sage advice on careers in the Hi-Tech industry (much of it valuable in any industry).
Your tastes will no doubt differ from mine - but the thing is, get in there and see for yourself - then if you want, make your own Podcasts and share them with the rest of us!