There are tons of these apps. I’ve only tried a few. Many of these apps can open and annotate pdf files also. I’ve tried a few others but these stand out. My current favorites are Noteshelf and Goodnotes.
Update: I’ve completely switched to Goodnotes. Combined with an Apple Pencil, this is just an amazing app. I often import meeting agendas into the app prior to the meeting (along with any other material) so that everything is in one spot.
Original Review:I purchased this after a friend recommended it. This app is good, but I prefer the Noteshelf interface. Goodnotes had a great backup function, however. I also like the way the writing space works. This app has some handwriting recognition also (limited right now). You can search on your notes. There is an output option to display on a larger screen.
Update: Still good, but I prefer Goodnotes now.
Original Review: This is an excellent app. The interface is terrific if a little dated. I’m not impressed with the decision to release in app purchases free (after a lot of people paid for them), but I like the app a lot. Unfortunately the only way to back up the notes is to use a computer and iTunes. You can export boys individually to the cloud as pdf files, but then you cannot restore them.
This app has one main advantage: you can record a lecture while taking notes. Very nice.
This is an Evernote product. It’s improved a lot lately. I like the scrolling magnified text entry area.