I've got an iPad that's got some important data stuck inside one of the apps, with no way to get it off (basically, it has a syncing problem with an external server). It's a third party app from the app store. I'd like to manually access the app's data, in the hope that I can extract it manually. How do I go about this using iTunes/XCode?
Install iExplorer, plug the iPad into your Mac and then use iExplorer to retrieve the data files.
Update: When I wrote this answer in 2013, this above was the case. However, Apple changed the security model in iOS 8.3 so that it's no longer possible for third party apps to access files in the app sandbox.
The makers of iExplorer have a detailed support page covering this issue.