Sharepoint Document Library
Create a New Document Library
Go to the site you want to add a library to (HR, Finance, Sales, etc.)
Select New
Under New, click the Document Library option

Select Blank Library

Give it a name then select Create

(Optional) If you leave the Show in Site Navigation option, it'll show up here

Library Access Permissions
Click the gear icon next to your name at the top right
Select the Library Settings option

Select More Library Settings

Select Permissions for this Document Library

Select Stop Inheriting Permissions

All Libraries, Lists, and content within a Sharepoint site have a parent inheritance flow of permissions. This means that whenever a Document Library or List is created, it's going to automatically inherit the permissions of the main site. Because we want to make this library private, we DO NOT want the same permissions that the rest of the site has. De-inheriting the permissions here allows us to make this Document Library's permissions unique.
Removing Members from Library
Select the Visitor or Guest User and Remove User Permissions

In my case specifically, I have the default settings. The default settings has a Visitors, Members, and Owners option. I have set my Sharepoint site to be viewable and accessible by Everyone within the IT Visitors group. Thus, if I do not want Everyone to be able to view the contents in this Document Library, I need to remove that Visitors group. You will need to make sure that you remove any group that you do not want to have access to the Document Library.
In fact, if it's easier, you can remove ALL of the groups and manually add the specific users that you want to be able to access this particular Document Library. Keep in mind that it's better to create groups such as "HR" and "IT" and "Finance" and add those groups specifically than it is to add individual users.
Adding Members to Library
Select the Members option or create a new group then select that group

Select New

Type in the user's name and select the result

Select Share

Access Document Library from Computer
Select Add Shortcut to OneDrive

It will show up under your OneDrive directory

This will work exactly like any other folder on your computer except it'll be on Sharepoint rather than the server.