Before we begin please note that you can not make uploaded files (pdf, word docs, images, etc) private, ever.
Even if you place an uploaded file on a "private" page the file itself is not private. 
If you upload any file to the site it will be viewable by the public and may be found by search engines such as google. Never upload any confidential file to the website.

Step 1

Click "edit" on the page you want to add a file to.  Once on the edit screen, click the "Files and Video" tab.

Step 2

Look for the "Add a new file" field.  Click "Choose File" or "Browse" (the language changes between Mac and Windows).  Locate the file on your computer, click "Open" and then click the "Upload" button.

Step 3

Your file should now display in the list of files, which is directly above the "Add a new file" field.  If you wish, you can add a description to your files.  This is the link text that will display on your page.  If you don't add a description, the file name will be the link text.

Step 4

Click the "Save" button, and you should see your files displayed at the bottom of your page/content.

Step 5

If you have multiple files on a page, and you want to change the order they display in, make your way back to the "Files and Video" tab of the edit screen.  Once there, use the quad-arrow cross to the left of the file name to drag and drop the files into a new order.  Click the "Save" button, and you should see your files displayed in the proper order at the bottom of your page/content.

Step 6

"But I don't want my files at the bottom of the page," you say (I usually say that, too).  To link to the file from anywhere on the page, right click (or control+click for mac users) on the file that you want to link to and choose "Copy Link Address" (for Firefox or Chrome) or "Copy shortcut" (for Internet Explorer). Then, make your way back to the edit screen for your page.

Step 7

In the body area of your page, highlight the text you want to use as your link text (see Add a Link for a refresher on adding links).  Click the "Insert/edit link" button, and paste the link into the "Link URL" field.  Save your changes, and admire the links you added.

Step 8

"Ok, ok, now my files are linked to from the proper spot, but they're still sitting at the bottom of my page, and I'm not a huge fan of that," you add (I usually add that, too).  You can fix that!  Make your way back to the edit mode for your page, and click on the "Images and Media" tab.  Next to your file(s), uncheck the box(es) in the "Display" column.  If no boxes are checked, the files won't display at the bottom of your page.  But they will still be accessible via those direct links you added in the body area.  Go ahead - check it out.

Step 9

Sit back and admire your work, you file-adding maven.