Apple's against file systems, so there method of sharing files between apps is more restricted than what you'll find in Android, but in short you should be able to download any file and share it to any app which (I believe) registers with iOS for supporting that file type.
For example, I went into Dropbox and opened the PDF they put in there, and after it downloaded in Dropbox fully, and opened, I was able to tap the bottom right "share" button, and it would let me open it in iBooks, or Evernote.