My opinion is that Apple already makes a camera, and it runs iOS.... However, in the spirit of "What If?"
Apple is about removing non-critical features, leaving the remaining features with enough capabilities for the user to be creative, imo. And they are good at connectivity and convergence.
So.... start with an iPod Touch as a base form factor. Apple won't make a camera with interchangeable lenses. Too many ways to mess things up and too many issues to support. However, the integrated lenses will have an optical zoom of about 24mm to 600mm (35mm camera equivalent)* and will be only average fast - say.... f/3.5. There will be iMage iStablization for low light shots.
Apple will do something revolutionary, for instance (hypothetically) it will use a cluster of 3 cheap-to-buy (for them) lense/sensor combos - essentially the same units used in iPhones
et al and then tied together with software to achieve something phenomenal - for instance (hypothetically) taking 3 low resolution sensors to achieve 28 megapixel files. Which by default will be saved as JPGs to save space.
The controls will consist of one rocker switch on top. Maybe it also slides so that you can lock the camera off, or lock the camera on and freeze the settings. Depress the rocker switch to snap the photo.
When you are taking photos, you aim the camera by looking at the screen, there is no dedicated viewfinder. The screen will be glossy, of course. The far right side of the screen, under your thumb, will be controls - and the GUI will be a touch enabled click-wheel.... you just move your thumb in circles to choose from Av, Tv, P, M. For the (P)rogram button you will have the choice of landscape/sports/portrait. The software will be smart enough for figure out when you want Macro, Night, etc. Even if you don't want that setting. The rocker switch will control the zoom/wide angle lense.
Apple won't actually use or show you Aperture values or Shutter speeds in Av and Tv, they'll use software to show you the picture, with the DoF and/or motion blur on the back screen. You adjust the photo, in real time before you take the photo, with the screen. Click the top-mounted rocker switch to take the photo. By default the camera will actually record a half or quarter second video clip. Which you can keep as-is, or you scroll through the clip and the pick the precise moment you want to record.
Once you have taken the photo it will automatically be GPS tagged, and the photo will start to upload to iPhoto on your host computer. There will be minimal storage on the the camera.....
However... The camera will keep the previews for all the photos you have taken in the past month, plus whatever you put into the camera's photo album. The previews will be compressed small, and perfectly viewable for the size of the screen plus 2x zoom. But.... you will be able to edit, email, upload to photo-sharing sites etc etc the photos using the same iOS photo apps now in use. But the apps are actually working on the full sized photos that are sitting on the host computer. The camera previews are showing the changes, but it's the photos that have been auto uploaded to the host computer that are being changed - using iCloud. And if the photo is being emailed, etc it will be sent from the host computer/iCloud.
You will be able to print and mail postcards/greeting cards from your camera. Maybe even calendars and books. Apple will buy the US postal service .... and make a profit running it to simply deliver the cards and calendars that people will start mailing to each other. (OK - that may be far-fetched

)
The camera will also run all of the iOS apps, but won't have telephone capabilities.
Sorry for the typos.... and the rambling-ness...