How do I make iTunes recognize a podcast downloaded from a source other than the iTunes store?

Sarah's advice works if you manually downloaded a file that you want iTunes to see as a podcast episode. But if you just want to ensure you have the latest episodes from a podcast you're already subscribed to, then read on. iTunes does not host podcast feeds or media. So if you used iTunes to subscribe to a podcast, then you're directly subscribed to the podcast's own RSS feed, not to anything from iTunes. So you depend on the podcaster to release their episodes, not for iTunes to update. However, iTunes will only check for new episodes every now and then. Go to your Podcasts section in iTunes and click the Refresh button to check all your subscriptions for new episodes. If you want iTunes to automatically check and download new episodes more often, click the Settings button, Settings for "Podcast Defaults," Check for new episodes: "Every hour," adjust the last two settings as you wish (I like "Download all" and keep "All unplayed episodes"). As long as iTunes is running and has access to the Internet, then it will check for and download any new episodes every hour, even if the Podcast's listing in the iTunes podcast directory doesn't show the new episodes. If you still aren't getting the latest episodes, contact the podcaster because they may have a problem with their RSS feed.

Add the podcast files to iTunes. Normally they'd be under the "Music" section, In order to make them appear in the "Podcasts" section: Right click the track -> Get Info -> Options -> Media Kind -> Podcast