Does the relative live close to you?
I use Audible and download the books to my iPod. However, there is a charge for Audible books. If you want to borrow free books from the library, your best bet is to go to your library and ask about it at their desk. Many libraries now have a system where you can "borrow" and "return" audiobooks on your computer without ever leaving home. Most of these are NOT compatible with an iPod; instead, you need an MP3 player. You can use headphones with any Apple or MP3 player; they all have a place to plug in the headphones. Your library will be able to tell you exactly what kind of player is required for their digital audiobooks.
Libraries still have compact discs you can borrow. If you live close to your relative and can physically go to the library to pick up and return compact discs, then that is an easy alternative for you if you do not want to learn how to use MP3 players, etc. For this option, in order to use headphones, you would probably want to purchase a personal CD player.
I would recommend making stops at both your local library and perhaps your local Best Buy for information. You need to decide whether to use digital audiobooks from the library or compact discs from the library. Your librarians may also be able to make recommendations for book selections for your relative.