"rooting" is, essentially, a device hack to attain unrestricted access to the entire file system within your cell phone. Sometimes manufacturers, like Apple for instance, will include restrictions to certain 3rd party apps - for whatever reason.
(Not sure, but I think the term "root" is used with Android products. I only say that because I have heard of "jailbreak" for Apple products. But they might be just interchangable slang.)
Rooting a phone means that you gain access to it as an administrator & can change systems and stuff that are usually read only. It can give you access to install certain applications and interfaces, but if done incorrectly, it will damage your phone (freeze it, etc.) and can make it unusable.
Rooting a phone yourself is likely to nullify your warranty, so check the terms & make sure you know what you are doing before proceeding.