Under the hood, an Android phone runs a modified LINUX operating system. Out of the box we, the end users, are given only certain permissions to perform certain actions on this system. We are locked out of other functions, largely for our own good, it is said. Manufacturers don't want us accidentally wiping our OS or allowing apps to do malicious damage.
Anyone who wants to get full control over their computer (I mean, phone) needs root access. This Gizmodo article lists specific reasons why "rooting" is a useful thing. My main reasons are to get full backup capabilities, prior to installing a new Android ROM.
One of the reasons I purchased a Jaiyu G2 is that it was in fact possible to find rooting information. This is not true of all Chinese phones. But the problem is that this information is fragmentary, confusing, and ill-written. Besides which, there are several different methods, requiring different levels of expertise and risk.
The best guide I found, by user "umit" on DroidChina, still lacks clarity for those of us new to this venture. So here I present a rewrite of that guide, all props to the original author.