Podcasting makes an audio file, available online for downloading via an automatic "feed." You can then hear the podcast whenever you want from your computer or a portable media player.
The first thing you need is podcast subscription software. Your options include the latest version of Apple's iTunes (which has the software built in), or other services such as Odeo or iPodder, to manage your podcast subscriptions. At this point, you will already be able to listen to podcast audio from your computer (provided your computer has a sound card and can play MP3 audio files). If you want to make your podcasts portable, you will need to transfer them from your computer to a portable media device.