A podcast is a series of downloadable audio files that you can subscribe to on your computer. Your computer regularly checks an online text file – called the podcast's feed – that lists any new episodes along with their download locations. Then the software downloads the new episodes and tees them up for you to listen to... on your computer, on an MP3 player such as an iPod (which is where the word "podcast" comes from), or however you like.
By far the most popular software for subscribing to podcasts is Apple's iTunes, although there are many other choices. (Here are some of them.)
One of the reasons podcasting has gathered so much attention is that it's a lot like radio, ... but unlike radio, it can be produced very cheaply. Nearly anyone with a microphone, a computer and an Internet connection can create a podcast, and there is a wide range of free and low-cost software available for producing them. Free services abound as well for uploading and hosting podcasts.
The same way blogging opened online text publishing to the masses, podcasting has turned audio publishing into a grassroots activity. More recently, podcasting has expanded to include video content as well in what are sometimes called vlogs, video blogs, video podcasts or vidcasts.
You can find out more at the podcasting resource library at Net2Learn. You can also check out the Wikipedia entry or the O'Reilly book Podcasting Hacks (a few years old, but still very good).