How to customize Windows 10 "6 Folders" in My Computer
The easiest way to completely customize the "Folders" in "This PC" would be to use the program "This PC Tweaker," available here: http://winaero.com/comment.php?comment.news.1836 (UPDATE (07/10/16): This software has been superseded by "Winaera Tweaker," found here: http://winaero.com/comment.php?comment.news.1836)
It was originally designed for Windows 8 and 8.1, but as of version 1.3 it supports Windows 10 as well.
This program will allow you to remove or modify existing folders, including changing names and icons, as well as add your own custom folders.
Below is the original answer, meant for just removing/restoring the original folders (which can also be done with the above program):
I had wondered this myself, and found the answer here: http://www.tenforums.com/tutorials/6015-pc-folders-add-remove-windows-10-a.html
Unfortunately there doesn't appear to be a way to do this through the GUI (as I believe was the case with Windows 8 as well), so we must take to the registry to accomplish this task. The link above provides .reg files that can be run to modify the registry for you. It has files to remove specific folders, or all folders, as well as files to restore them. Make sure to choose the correct one for your OS version (32-bit or 64-bit).
If you would like to check the contents of the files, they are in plain text so it is easy to verify that they are not malicious.
Before making any changes to the registry, I would recommend making a backup of the registry. To do this, open the registry by typing "regedit" (without the quotes) into the start menu. Select "Computer" at the top of the list on the left, then click File>Export. Save wherever you would like. This way if the registry changes cause problems, you can easily revert to a working state.
Windows 10 (32-bit)
; Music
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{3dfdf296-dbec-4fb4-81d1-6a3438bcf4de}]
; Downloads
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{088e3905-0323-4b02-9826-5d99428e115f}]
; Pictures
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{24ad3ad4-a569-4530-98e1-ab02f9417aa8}]
; Videos
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{f86fa3ab-70d2-4fc7-9c99-fcbf05467f3a}]
; Documents
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{d3162b92-9365-467a-956b-92703aca08af}]
; Desktop
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{B4BFCC3A-DB2C-424C-B029-7FE99A87C641}]
Windows 10 (64-bit)
; Music
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{3dfdf296-dbec-4fb4-81d1-6a3438bcf4de}]
[-HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{3dfdf296-dbec-4fb4-81d1-6a3438bcf4de}]
; Downloads
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{088e3905-0323-4b02-9826-5d99428e115f}]
[-HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{088e3905-0323-4b02-9826-5d99428e115f}]
; Pictures
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{24ad3ad4-a569-4530-98e1-ab02f9417aa8}]
[-HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{24ad3ad4-a569-4530-98e1-ab02f9417aa8}]
; Videos
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{f86fa3ab-70d2-4fc7-9c99-fcbf05467f3a}]
[-HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{f86fa3ab-70d2-4fc7-9c99-fcbf05467f3a}]
; Documents
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{d3162b92-9365-467a-956b-92703aca08af}]
[-HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{d3162b92-9365-467a-956b-92703aca08af}]
; Desktop
[-HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{B4BFCC3A-DB2C-424C-B029-7FE99A87C641}]
[-HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace\{B4BFCC3A-DB2C-424C-B029-7FE99A87C641}]