When you subscribe to a new feed it comes with (sometimes long) list of available episodes. Based on your preference you can either download an episode, or play it directly without downloading (also known as "streaming"). Once episode is downloaded (by using the "Download" button), the episode content is stored locally on your SD card and you can play it regardless if you are online or offline.
You can also play (or add to the playlist) an episode without downloading it first. In this case BeyondPod creates "streamable" episode. Streamable episodes are similar to file shortcuts - they behave the same as any other episodes, except that they don't take any space on your device. When played, episode content is downloaded "on demand" so you will need an Internet connection to play them. You can distinguish streamed episodes by the green ((STREAM)) icon in the bottom left of the episode.
Both Downloaded and Streamed episodes are often referred to as "My Episodes" because they are episodes that you have expressed some interest.