Amazon's newest media service is Prime Music, a music-streaming subscription that's included with Amazon Prime. Though it's not quite at the level of Spotify or Rdio, it has a music catalog of one million songs, so you're sure to find something you like.
All of the music that has the "prime" label means you can stream it for free. The Prime Music section of the app is a bit confusing to browse, but you can use the handy search feature to hunt for your favorite artists and songs.
In Prime Music, you can view individual albums and add the entire tracklist to your Library at once. You must add tracks to your Library before you can play them in the app, and you do so by tapping the "Add" button.
When you tap a song in your Library, it will start playing, bringing up this playback screen with controls at the bottom and the album art. You can tap those two arrows at the top left to collapse this view, so you can continue to browse the app while the music plays.
Amazon "music editors" created thousands of playlists for Prime Music, based on artists, genres, and moods. If you're unsure if you'll like a particular playlist, you can sample it, which plays 30-second clips of every song.