iFile
The best way that I can describe iFile is that it's like File Manager for your iPhone. It allows you to easily browse system files, app files, and more, all from an easy to navigate application interface.
When you first open the app, you'll see a little gear icon used to manage your preferences, a Wi-Fi icon used to start your web server, a bookmark icon, a home button, and an information tab. From home, you can browse the /var/mobile directory and beyond.
I really like that you can setup a web server for transferring files and/or testing more advanced app functions. This is great for transferring information when you don't have a cable handy.
From the preferences menu, you have the option of things like Hidden Files, using a trash can, browsing by song name and/or application name vs file name, as well as thumbnails. My only gripe, if there is one, is that only registered (paid) users get access to most of these features, in addition to search functionality.
Like many good file managers, iFile allows you to quick view files, play audio, rename files, and even edit on the fly. It supports bluetooth transfer, database view, and even Microsoft File types like doc, xls, and ppt.
Though I'm not one to mess around under the hood as much as others may be, I do find that having a utility like this is great for downloading app resources as I try to learn more about how an app might be built or how the iOS file structure operates. That being said, for someone like me, that's where the tinkering stops.
Still, being able to transfer files via web server is worth the download alone, especially since this app is free.