For the most part documents stay in the app that created them. If you download a document in Safari or the Mail app, you need to copy that file to a compatible app. There is no folder where they just automatically show up.
For instance, if you tap on a PDF file in Safari it will download and display that file. When you are viewing the file there will be Open In options at the top right of the screen. This tends to go away (so as not to clutter you view) but you can tap the center of the screen to bring it back.
When you choose the generic Open In, you'll be presented with a Share Sheet; a list of compatible apps and actions. Choose one and that's where the file goes. In some cases Apple may present you with a shortcut "Open In XXX". How it chooses this app is a mystery, but if there is an Apple app available, it's almost always that. iBooks always shows up for ePub files, and almost always for PDF.
If Safari can not view the file type, it should give you Open In options after you tap on the link and the file has downloaded. If there are no compatible apps, you may not get any options. It's a good idea to have at least one app that can handle general file management in that case. I like GoodReader.
Mail is similar. If you open the file to view, you'll get the same reading / saving options.
If the file is displayed in-line (most pictures and one page PDF's are), then tap and hold in the center of the item for the Share Sheet.
If a file has not been downloaded you may have to tap on it to start the download. Once downloaded it will either display in-line, or you can tap on the icon for the same share sheet.
Third party apps may let you save files more directly. But if they do then they are saving those file in that app.
There is also Cloud Drive. If you activate it then the folders for enabled apps will become synced with iCloud Drive. Sometimes, not always, it is possible to select iCloud Drive as a destination. In those cases iOS acts a little bit like a normal File Management system; but there are distinct limitations and a lack of management features. Still, it comes in handy for shuffling files form one app to another without having to make a copy for each.
iCloud Drive counts against your iCloud storage limit. So keep that in mind.
Most of this is covered to some extent in the iPad User Guide for iOS 9. You can find it as a PDF on Apple's site, or get it for free in the iBooks store. I recommend the later, as it's much easier to search.
