Podcasts are audio shows with individual episodes, just like a radio show with different episodes. Podcasts are made for listening on your iPhone or Android phone, though you can also listen to them on the web or other places.
If you are listening on your phone, the best way to listen to an episode is with an app. There are many different apps that basically do the same thing. You can use them to find the podcast you want. If you like it, then you can "subscribe" to it, and the app will automatically download new episodes.
iPhones come with an app called "Podcasts." Other popular apps are Stitcher and Overcast. They look like this:
Android phones come with the Google Play Music app that can be used to listen to podcasts. Other popular apps are PocketCasts and Stitcher.
This video explains how to listen to podcasts, brought to you by Ira Glass from This American Life.