RadarBox24 (free) is a slick iOS app that displays a zoomable map showing airlines flights that are currently en route. Zoom in and you see representations of the planes moving along their routes. The aircraft location information comes from the servers at AirNav Systems, and is the same data used by airlines and airports. In addition to the United States, the app also provides coverage for about 90 percent of Europe.
The free version is pretty basic, showing all the flights but also displaying ads. Your map can be a terrain or plain map, or the app can display a satellite view.
A US$2.99 upgrade to the Pro version adds a sky view so you can point your phone at the sky and identify flights as you watch them in real time with your own eyes. I tried the sky view and it worked well, but the data just slightly lagged behind what I was seeing. That's really not a surprise given the complexity of what the app is doing and the speed of the data stream.
Both versions of the app are impressive. I upgraded to the Pro version and liked the added information I could get. There is also a search mode for particular flights and airports. Clicking on an aircraft button provides more information about the flight, like its altitude and speed, as well as a photo of the aircraft type. In my opinion, the Pro version is reasonably priced.