What are podcasts?
Podcasts are basically audio (and sometimes video) files that you can subscribe to. The actual podcasts themselves take on may different formats. Some are produced like high-quality radio shows, some are basement productions about local events or interests, others consist of music and still others regard just about anything you can imagine, such as language learning.
The key here is that you can subscribe to these audio and video "feeds" so that when a new one comes out, you can be automagically notified in a number of different ways. The way that you are notified depends a lot on how you subscribe. There are many types of "RSS readers" and "News Aggregators" that can aid you with this subscription process. (See links at bottom for more info about these.)
Tools for Subscribing
Here are some links to lists of available RSS readers for Mac and Windows. (Linux? BSD? You know what you are doing!)
For those who want a good one to run with, I would recommend these Free Software RSS Feed Readers:
Many web browsers such as Firefox can manage RSS feeds for you.
Free Language RSS Feeds
Free Language has numerous RSS feeds you can subscribe to for automagic updates. Every language has an RSS feed, and these show up in the left sidebar. Just click them to subscribe.