In order to listen to or view podcasts, you need a so-called "podcatcher", which enables you to download, subscribe and play back podcasts.
There are various podcatching solutions, such as:
Podcast software (such as "iTunes" or "Zune") which enables you to either load podcasts to your portable player or watch or listen to them on your computer.
Mobile devices, which support downloading, subscribing and playback of podcasts directly from the internet (to watch video podcasts, you need a mobile device offering video functionality).
Online services supporting the integration of podcast to a personalised home page (such as e.g. "myAOL").