Activities provide a rapid entry method to capture and manage short, simple pieces of work that don't require the full overhead of a project.

Once an Activity is created the owner can edit or delete it regardless of their other security permissions. So users who need be to able to create and manage their own Activities only need the Activity: Create permission.

Users who need to be able to view, modify or delete ANY Activity need Activity: View, Activity: Modify and Activity: Delete. (These permissions are on the Business Alignment tab of the Organisation Security Roles page).