Linux / Folder and /root folder

/ (Root directory) vs /root directory

1. / directory called as Root Directory sits on the top of the file system hierarchy.
   • That means it is the ultimate parent or grandparent or grand grandparent of any file or directory you can find on your system.
   • It contains all the files necessary for the system to work such as boot files, libraries, packages, essential binaries, system configuration, user files and temporary files.

_source

2. Now /root is the Home directory for User named Root. See at the bottom right in above image.

   • Just like every User has it's own directory with his/her username under /home, User Root must have also a directory.
   • But Since Root user needs to know every tiny detail about system, so his home directory is created under / itself by the name /root
   • It contains the files and folders you created when you were root user and also the hidden configuration files for some applications or packages you installed.

Now when I sudo cd root/ I stay in the same folder (/)

As pointed by cylglad in the comments,

cd is a builtin shell command, so doing sudo cd /root won't work

See this :

amit@C0deDaedalus:~$ cd /root
bash: cd: /root: Permission denied
amit@C0deDaedalus:~$ 
amit@C0deDaedalus:~$ sudo cd /root
[sudo] password for amit: 
sudo: cd: command not found

Instead you have to first change to root user, then do a cd to /root

$ sudo -i
# cd /root
# pwd

Feel free to add in more details.