Pandora® internet radio It's so easy to use and personalize. I love the option of thumbing up or down certain tracks as well as the ability to "seed" with other artists to focus in the station to my likes.
Spotify Not only does this app play streaming radio based on artist or genre, but you can also play just a single artist or album. Plus it pulls from your itunes library as well and can sync with your desktop application.
8tracks radio A very cool alternative to pandora and some of the more popular music apps. 8track offers a selection of curated stations so your music is chosen by real people, not an algorithm. Only thing that makes it falter is it often drops off connection.
Stitcher Radio - News & Talk Sometimes I enjoy news or sports talk radio, and Stitcher is great for that. I can download podcasts and love the feature of being able to "stitch" together shows to make my own station.
Rhapsody This is a really powerful app, but keep in mind it is subscription based. But what you get with the monthly fee is unlimited music and access to pretty much any songs imaginable. You don't have the limitations of some of the free apps.