Suppose you want to allow a specific group of users to access particular categories, while forbidding everyone else from those categories.

For example purposes, we'll say the people you want to allow are called 'customers'. Create a 'customers' usergroup at Admin Panel -> Members -> Manage Usergroups (just use the same permissions as regular members unless you want different). Move the desired members to that usergroup by editing their profiles. Then, edit the specific categories you have in mind (either through Admin Panel -> Categories -> Bulk Edit or visiting the category and clicking edit in the editorial controls area). On the edit category page, there's a "Display category permissions for editing" link near the bottom. Click that to expand that area. Then select "no" for "Can view category?" for the guest and member usergroups, "yes" (or leaving as the default would work too) for the customers group. You now have a category that can only be viewed by customers.