A Podcast is an online audio or video production. What makes it different than any file you download on the web is that it has the ability to be downloaded automatically. It uses RSS (Really Simple Syndication) to accomplish this. Podcasts are downloaded to a personal computer or portable player.
Contrary to the name there is no need for an iPod to listen to podcasts. All you need is a pod-catcher. There are many of these available, they automatically fetch podcasts when publishers add new episodes. Two examples of popular pod-catchers are iTunes and Pocket Casts (iOS and Android). If you prefer, you can also synchronize your podcasts with various web services, a popular one is Podnova. There is no need though to use one of these clients or services if you don’t want to. You can simply download the MP3 just like any other file online.
You can check out Wikipedia for a more in depth article on podcasting.