If you follow certain video or audio podcasts regularly, you can use Apple's Podcasts app to do so. For podcasts you listen to or watch on a regular basis, it's typically better to just subscribe to them. If there's ever a time when you want to download a single podcast episode without subscribing, you have the option to easily do so.
Launch the Podcasts app from the Home screen of your iPhone or iPad.
Tap on the Store icon in the upper left-hand corner.
Search for the podcast that you'd like to download an episode of.
You can either tap the download button next to the single item you'd like to download under the Episodes section or tap on the actual podcast under the Podcasts section.
If you're in the actual podcast screen, you can scroll through and view descriptions and summaries of each episode and tap the Download button next to only the ones you'd like to download without subscribing.