Getting root or rooting your phone is the process of modifying the operating system that shipped with your device to grant you complete control over it.
This means you can overcome limitations that the carriers and manufacturers put on your phone, extend system functionality, and even upgrade it to a custom flavor of Android.
The name root comes from the Linux operating system world, where the most privileged user on the system (otherwise known as Administrator on Windows) is called root.
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rooting does *not* mean installing a custom ROM (a ROM is a modified, "aftermarket" OS).
Installing a ROM may require rooting first, but just rooting can be usually done in only a few minutes, keeping your stock OS otherwise completely intact.