Apple recently released a Podcasts app for iOS. This application has a number of performance and interface issues, but some podcast listeners may find it superior to using the Music app (though many standalone podcast apps have far more features than Apple's app).
When you look at the Podcasts app, it seems as though you can only subscribe to podcasts that are in Apple's store. This is not true, but the way to subscribe is unintuitive.
When you're in the Podcasts app and viewing your podcasts, there's a black bar at the bottom of the window with a Search field to the right. Paste a podcast URL into this field and tap on Search. A dialog will display asking if you want to subscribe to the podcast. Tap Subscribe to add it to your podcasts.
Mac OS X Hints