Podcasting is a general term for any sort of web-related, episodic audio content or radio that can be streamed or downloaded to computers, mobile devices, or smartphones. In many ways, a Podcast is like an audio version of a blog. There are thousands of podcasts currently being broadcast from around the world and pertain to just about any subject imaginable. Many radio programs are available as podcasts after the show has aired.
Whether you are going to listen to a Podcast on a device or on your computer, you will need to use a particular type of app called a "podcatcher." These programs allow you to subscribe to a podcast and will automatically aggregate and download episodes to your device. Here are some free programs for popular platforms:
If you use Apple products and iTunes already, this is a non-brainer. Search online for a podcast an paste the URL into the "Subscribe to Podcast" option under Advanced. You can also use iTunes to search for other casts.
iTunes is freely for Mac and PC and can be used to subscribe to podcasts.
This American Life
The Splendid Table