Android is a compact, real-time Linux.
The name of the admin user in Linux is root - so rooting is the exact same thing as having Administrator privileges on your PC.
Right now, only your phone manufacturer and carrier have such rights.
The act of rooting installs a permission-management service library and a corresponding control app to the otherwise protected /system area. The two together are generically called superuser, but the app you see may have a different (but similar) name.
Once you have that, you can attempt to perform restricted or admin actions - the attempt gets stopped by the superuser process and you get a pop-up alert to confirm your intention.
You may hear people say that the name comes from getting access to the root filesystem folder - or /.
Access to the lowest level comes from an unlocked phone, with superuser granting access to remove read-only permission on the /system or other areas.
It's a popular myth that root means root file access and no one will argue because it sounds good.
But - you have root access to sd card and that doesn't make you the admin ok.
So yes - rooting changes your protected /system area by adding two files, with nothing else changed - until you change it.
If your phone's root forum here lacks instructions on how to replace your stock recovery (the thing that recovers carrier OTA update files and installs them) with a custom recovery then that probably means that no custom recovery is available for your phone.
Without custom recovery, you can't make image backups to save you from any mistakes you make playing under the hood - so you have to proceed with caution no matter how much Internet advice you see on how easy or great some mod is.
Root gives you great power and can make your phone ten times as fun. I do it myself.
It also gives you the power to convert your phone into a useless paperweight.
How to root varies by phone model.
Some phones are root friendly and follow the standard Android rooting model -
Unlock the bootloader.
Install a custom recovery.
Use the custom recovery to install superuser.
Other phones aren't so lucky.
In some cases you get root by running various programs or techniques to sneak in the superuser files to the protected system area (that is designed against that) through a split-second timing exploit or security weakness.
And on some, you can't get root at all.
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The point of rooting is personal control.
You can disable unwanted carrier bloatware.
You can install a proper firewall and an ad blocker.
You can add convenience enhancements. (For example my phone has the headphone port stupidly on the bottom of the phone. OK, I have a root mod that lets me rotate my screen 360° - presto - I turn my phone so the headphone jack is on top and the screen isn't upside down.)
You can add control enhancements. Google says that the right number of buttons on the bottom is 3 - I say it's 5.
And the number one reason to root, no matter what else you do or don't -
Backups.
With root you can backup your entire phone.
Without it, you can't.