SkyView Free
This app blends the iPhone's camera with 3-D imagery of the night sky to show you what's up there, complete with information about each star, planet and celestial object in its database. It displays the current position of the International Space Station (ISS) and other satellites. If you're looking for a specific object, the search feature lets you follow the arrow to find it in the sky. Not just for use at night or outdoors, SkyView Free works any time, anywhere -- during the day and inside, too. Best of all, as the name implies, it's free.
Pocket Universe: Virtual Sky Astronomy
Pocket Universe also gives you a fantastic augmented reality view of the night sky, as well as giving you "Tonight's Sky" and "Objects and Events" options to notify you of particularly interesting things to look for as you head out for the night. It can provide you with notifications for when the ISS will be visible from your location, link to Wikipedia for more information about the object you're looking at and quiz you on stars and constellations to help you learn the skies above you. $2.99.
Luminos -- Astronomy for iOS
Luminos gives you even more detail from your iPhone stargazing experience. Featuring eclipse maps, advanced satellite orbit data and detailed 3-D renders of objects with the capability to make notes about what you're viewing, this is a truly powerful and beautiful app. Each object's full coordinates are included, and you can even connect your phone to your telescope and control it directly from the app. Packed with more than 2.5 million stars in its database, there's more than enough to keep you busy. $6.99.
SkySafari 3 Plus
If you've always wanted to go into space but never wanted to go through all that pesky astronaut training, SkySafari is for you. Featuring 2.5 million stars; 31,000 deep sky objects; 4,000 asteroids, comets and satellites; and the entire NGC/IC catalog, if it's in the night sky, this app can show it to you. The app also features NASA imagery of planets, moons and celestial objects taken with the Hubble. Interactive star charts and updates on all major sky events are included, and it can serve as a telescope controller too -- just tap and go. $14.99.
Disclaimer
Information in this article applies to SkyView Free v2.1.1, Pocket Universe v4.7.8, Luminos v7.2 and SkySafari v3.7.3 for the Apple iPhone. It may vary slightly or significantly with other versions or products.
Prices are as of June 2013.