This is probably one of the best radio apps, even though it's not a "true" radio app. Like iheartradio, it has a large catalog of songs, but instead of porting live radio stations, it generates a custom station tuned to your interests. Simply enter a favorite song or artist and Pandora will provide you with that music, as well as songs that are musically similar.
How does it do this? Pandora has employed a number of musicians to sit down and categorize all the music they can get their hands on. The process has more to do with music, style, and technical elements than opinion or quality. If you put in Pat Benatar, for instance, you'll probably receive more songs with a strong female vocal lead. From there you can discover both popular bands and hidden gems.
Beyond the service, the Pandora app itself is great. You can skip forward a reasonable number of times in one sitting and you can rate the songs with a thumbs up or thumbs down to help tune your station. Like iheartradio, Pandora has background play enabled, so you can surf the web and listen to some great tunes at the same time!
The app IS ad-supported, but it's not very intrusive. If you go in for the "Pandora One" subscription (36 bucks a year), you'll have greater freedom to skip songs and you'll have no ads (this especially important on the web version, thought the benefit carries over to your iPhone app).
The app and the service are always available free, though, so check it out on the link below!