Taking a panoramic photo with iOS 6 is simple. Launch the Camera app and tap on the Options button along the top of the screen. Select the panorama button.
You'll then see some instructions on the screen. From there, it's easy. Move your phone in a slow, steady motion from left to right, making sure the arrow stays centered on the line. If you begin to move too fast or get too far off center you'll see a message on the screen telling you how to correct it.
The key here is to go slow. In my testing, even if I was going slow enough to avoid prompting the "Slow Down" warning from my phone, the photo often came out blurry. The slower I moved, the better the end product looked -- so take your time.
If you don't want to take a full 240-degree photo, you can stop at any time by pressing the shutter button along the bottom. Whatever progress you have made will then be saved to your Camera Roll.
Of course, taking a panoramic photo with your iPhone has long been possible thanks to third-party apps in the App Store. It's nice to see Apple roll out a feature of its own, but it's too early to say whether or not this is the best solution.