Camera apps seem to get most of the attention from iPhone users, probably because there are a lot more people taking stills than video. The iPhone does have very capable video features, and some third parties seem to have outdone Apple when it comes to creating and editing videos you will want to share.
Videon is a perfect example. This US$2.99 app lets you shoot video as well as edit out those bits that should be left on the virtual cutting room floor. Videon's feature set is complete, with a nice-looking digital zoom, multiple resolutions and aspect ratios, adjustable frame rates for slow motion, independent focus and exposure points, and the ability to take a photo while you are recording a video.
Once you have captured your clips, Videon is an excellent video editor. It provides effects like saturation, filters, black and white conversion, and even the ability to straighten (or tilt) an image that isn't oriented the way you want. Clips can be merged into longer videos and re-ordered, as well as trimmed.
I gave Videon a spin and found it both reliable and intuitive. I thought the digital zoom was of very good quality, but it is no substitute for a real zoom lens, something Apple doesn't supply (Editor's note: although the company has added a digital zoom to the iOS 7 beta). Still, the zoom Videon provides is of the best quality I have seen on an iOS app. Stabilization was also quite good, but if you don't have steady hands understand that your frames will be cropped in order for the feature to work. Stills can be grabbed from videos after they are taken, and the final video saved to your camera roll.
Videon is an impressive app, and goes far beyond what Apple builds into the iPhone. I would consider Videon to be among the top consumer-oriented video apps on the platform. Those with a more professional bent should take a look at FiLMiC Pro for $4.99. which is aimed more at pros using the iPhone camera. The ultimate app for video appears to be Vizzywig, a camera and editing app that even includes titles, music and the ability to share video with others running the app in real time. I'll try to get my hands on it to give it a test. User reviews are very positive, and Vizzywig is $19.99.
There's a YouTube video online if you want to get a good overview of the Videon features.
Videon is optimized for the iPhone 5 and requires iOS 6 or later.