The following was written by forum member 'AmazonAddict' and all credit should be given to him for this great explanation:
I'll give it a go to explain what 'rooting' is. When you hear the term 'root' or 'rooting' what it in essence means is that you've modified the tablet software in a way to gain 'super user' or highest level administrator status. In other words, any limitations of what can and can't be used on your device, what websites or software can or cannot be run...anything that the manufacturer did to 'limit' users on the normal 'unrooted' device can now be overcome or undone.
Now once a device is 'rooted' it let's you do a lot of things that you might not otherwise be able to do. Each device is different and each manufacturer puts their own limitations, custom interfaces etc on their device. With the Kindle Fire users are normally prevented from purchasing Apps and media from non-Amazon sites. Of course this is smart business for them since it keeps the money on their site. But for Kindle Fire owners what if you would rather choose Apps from a different site like the 'Android Market' or from other sites as well. Well in that case you might not appreciate Amazon limiting your access. Once rooted you can surf to any site and download from any site you want. It's up to you.
Now you might also hear talk of 'custom Roms'. When you hear this what they are referring to is usually a different operating system that is installed onto your tablet either in place of...or in addition to the OS that comes from the factory. For instance there is a very popular custom Rom called CyanogenMod that is a full version of Android 2.3 (Gingerbread) that let's you use any Android software you want and customize to your hearts content like add wallpapers, widgets and all sorts of fun stuff...that the stock Kindle Fire OS would never let you do.
Some people even have both...which is typically called 'dual boot'. What this means is that when you start your device you're given a choice of which OS to load up. You can choose the original Operating system...or a custom one like CyanogenMod. Right now CyanogenMod7 is the version that is out and it is Android 2.3 but they are already working on a new version called CyanogenMod9 that is a version of the brand new Android 4.0 (Ice Cream Sandwich) which when available will let those who choose to root...put the latest and greatest Android OS onto their Kindle Fires.
Now one more word on 'rooting'. Because you are changing the operating system of your device anyone that chooses to 'root' their tablet and especially if putting a custom Rom on should realize that very likely the company, in this case Amazon might consider it voiding your warranty. If something should happen during the process...even if unlikely...it would be your responsibility to realize you might be stuck with a paperweight instead of a working tablet . Now usually if you follow the instructions that are given and since so many people do it, it is usually a pretty smooth process and 95% of the time nothing goes wrong and that's the end of the story. Just make sure you understand there is always some risk. make sure you read instructions and forum posts etc and follow things exactly if you end up rooting at some point. I've done it to several devices and never had any problems but you always hear of horror stories nonetheless.
I hope that made sense. Let me know if you have any questions.