My favorite productivity app, by far, has been Notability ($2.99).  I primarily use it to jot notes onto pdf documents during meetings, to provide handwritten feedback on graded assignments, or to create handwritten solutions with sketched figures and equations.  There is a text/type tool as well, and a voice recorder.  You can add photos,screenshots, and webclips to a document, or turn on/off lined paper in the background.  The highlighter and eraser tools are intuitive; the scissors tool lets you cut and move your work by drawing a circle around it and dragging.  Two-finger gestures allow you to pinch, zoom, drag and rotate. It allows importing/exporting/syncing with Dropbox, GoogleDocs, iCloud, and Box accounts.  It will import pdfs as well as Office files (.docx, .xlsx, .pptx) and export files as either pdf or its native Note format.  You can send files to Notability from within other apps.  Within the app, you can organize your documents into Subjects (folders) and Dividers (groups of folders) and choose the color and icon for each Subject.

Tip for rookies:  TWO fingers to scroll, ONE finger to write.  I still struggle with this — thankfully the “undo” button is prominent!

Similar FREE apps that I’ve tried:  Educreations (good for recording screencasts), Evernote (is trying to take over the world?), Paperport Notes, 53 Paper (free app to go with an expensive stylus (that I don’t own)), and Doceri (designed to be projected).

Similar not-free apps I’d like to try, based on reviews by academics:  GoodNotes, PDF Expert, iAnnotate.  If you’ve used any of these + Notability, let me know how they compare!