Profiles let you group calendars when working with multiple calendars. If you do not yet use multiple calendars, please first read the article Organize your events in multiple calendars.
By default you are in the ALL profile, a static profile containing all calendars (except calendars deactivated in the calendar list which are never shown anywhere).
aCalendar also creates static profiles for each of your cloud accounts and for local calendars.
If you switch to any other profile, you will see the profile icon instead of the hamburger menu icon.
In widgets you can select a profile in the widget settings and also whether the app should open in the widget's profile or in the default profile.
To switch profiles you can always open the menu and click on the profile selector at the very top.
Alternatively you can configure the volume buttons, 3-finger-tap or tap on the date in the title to switch profiles (all configurable in settings - gestures & buttons).
Managing profiles (requires aCalendar+ or business features)
To manage profiles open the profile selector in the menu and click on manage profiles.
There you can add profiles with the plus button in the bottom right corner.
To edit/delete profiles or set a profile as default, open the context menu of the profile (by long pressing).
To quickly see one (or more) calendars without creating a real profile, there is the Instant profile option in the profile switcher.