Activity notes are actually added to contacts but are shown in both Company view and Contact view.

When you are adding an Activity Note you are recording the fact that an activity took place with someone at a company -- e.g. a phone call, an email, a meeting outcome.  
To add an activity note from the Company view you need to click on the menu button next to one of the contacts that belongs to that company 

When in Contact view you have two options to add an activity note.  You can either use the menu on the left hand side of the screen or you can click on the green + sign in the activity shortlist on the right hand side of the screen.