To illustrate my question or my answer I usually add a screen shot (like the one below) containing parts of the iOS or OS X operating system.
Do I violate Apple's copyrights when doing so?
And if I do violate Apple's copyrights, what is Stack Exchange's policy on this?
Copyright depends on many elements - some contradictory.
The content policy is linked from the image uploader window:
The policy clearly anticipates accepting users posting material encumbered with copyright:
Using copyrighted material does not constitute infringement in all cases. In general, however, users should be careful when using copyrighted content without the permission of those who created it.
The first is jurisdiction. Where are you subject to laws and where was the agreement (if any) established between you and Apple. A general statement is as pointless to make as "Who would win in a fight between a Gorilla and a Shark?"
As to policy, you can find Stack Exchange's submissions licensing requirements at the legal link towards the bottom of any page here.
When you upload an image, you are posting a link to that image - which to the best of my knowledge is stored, hosted and otherwise located at imgur.com and not actually on the Stack Exchange servers.
If you wanted to narrow down say existing copyright case law and decisions in the USA between US companies and US citizens - there might be some interesting interplay between Apple's EULA for a specific OS version and settled cases.
That question would be better on the main site as we already have some people asking if you can use photographs of Apple Products in your movies and in this sense - you are making original art by deciding how to arrange, crop, present and add narrative to what Apple's software can make on your screen. It would be odd for someone to be prevented from making art with a musical instrument or computer and software that they purchased for that express purpose. The musical instrument is created to make noise just as computers are created to make noise, pictures and heat.
Since you are not passing off Apple's design elements as yours, you are not competing with them or using their work in your product, it's hard to see how US copyright protections that Apple has rights to are infringed by licensed users of OS X posting screen shots of what they are seeing on their computer screens.
To add to bmike's execellent post:
Fair Use provides most of the exemption coverage for screenshots on the SE network. Because this is a Q&A site, academic in its very nature, use of copyrighted material is generally fine.
