Evernote is my go-to app for note taking, organizing files and clipping things off of the web. It is my main organizational tool. It works for me. I also using Google Keep for a lot of notes.
Not everyone likes it or wants to have another company have a hold of all of their data. That's where Paperwork comes in. It isn't as full featured as Evernote, but you can host it yourself and it does have some great features. You use notes, notebooks, tags and can upload multimedia files. There are not any mobile apps, yet, but you can access it on any web enabled device
Schools could use this to host their own note taking system, or as a tool for students to learn and manage themselves.
There is more info about it here on GitHub.
Evernote for Education Resources
Tips to Using Google Keep in Education
Lots of Great NoteTaking Apps and Resources