Why do playlists I create in the iPod app disappear? Every playlist I have made in the iPod app has at some point vanished. What causes this and how do I make them persist?
When you create playlists on an iOS device, those playlists get added to your iTunes Library whenever you sync. Once this happens, the fact that they were created on the iOS device doesn't matter anymore as far as iTunes is concerned--they're just more playlists that need to be synced back and forth according to your sync settings for said device. 

On iTunes, plug in and then select your device in the Devices section. Then, in the Music tab make sure that you have enabled the option to select individual playlists. As long as this option is enabled, you will be able to select and deselect individual playlists. Make sure any new playlists are selected there as well. 

Another (less common) scenario is when your iOS device is running a much older version of iOS. Previous versions (back when it was still called iPhone OS) had the concept of "On The Go" playlists as it existed on the original iPod. If your device still has "On The Go" playlists, then the behavior is much different: Those playlists get added to your iTunes library as "On-the-go 1", "On-the-go 2", etc; on which case they may not be synced up if they're not re-selected on sync screen (Music tab) for your device. The original iPhone and iPod Touch still had "On The Go" playlists. 

Those are the two scenarios I know of that would lead to playlists not being synced back. Could you elaborate on what your device type and iOS version is? That may help us troubleshoot your problem better. Hope this helps.