A podcast is a collection of online audio or video content that can be subscribed to using a Podcast catcher application via the RSS protocol. Video podcasts -- also called videocasts, vidcasts and vodcasts -- combine the audio component of podcasting with visual media.

Once you subscribe to a Podcast all future content will be automatically downloaded as soon as they are available.

iTunes

The most common way to subscribe to Podcasts on a computer (PC or Mac) is to use iTunes. 
Click the icon to install it. Once installed click Advanced -> Subscribe to Podcast... and enter in the RSS feed URL as shown below.

iOS App Store

There are many Podcast catchers available for iOS devices, however one of the best is Downcast ($2.99 in the app store). 
Click the icon to install it. Once installed, launch Downcast, click the + icon, and manually add a feed with the RSS feed URL as shown below.

Google Play

There are many Podcast catchers available for Android, however one of the best is DoggCatcher ($2.99 in Google Play). 
Click the icon to install it. Once installed, launch DoggCatcher and manually add a feed with the RSS feed URL as shown below.